:: Darboux's Theorem
:: by Noboru Endou , Katsumi Wasaki and Yasunari Shidama
::
:: Received December 7, 1999
:: Copyright (c) 1999-2026 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies FINSEQ_1, ARYTM_1, INTEGRA1, RELAT_1, ORDINAL2, FUNCT_1, CARD_1,
FUNCT_3, SEQ_2, JORDAN3, RFINSEQ, INTEGRA2, FDIFF_1, ARYTM_3, SEQ_4,
CLASSES1, VALUED_0, XBOOLE_0, NUMBERS, MEASURE7, REAL_1, SUBSET_1,
XXREAL_2, XXREAL_0, NAT_1, TARSKI, ORDINAL4, COMPLEX1, CARD_3, XXREAL_1,
PARTFUN1, SEQ_1, MEASURE5, FUNCT_7, XCMPLX_0;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, CARD_1, NUMBERS, XXREAL_0,
XREAL_0, XCMPLX_0, COMPLEX1, REAL_1, NAT_1, NAT_D, RELAT_1, FUNCT_1,
FUNCT_2, XXREAL_2, SEQ_4, VALUED_0, PARTFUN1, MEASURE6, FINSEQ_1,
RFUNCT_1, RVSUM_1, INTEGRA1, SEQ_1, COMSEQ_2, SEQ_2, FINSEQ_6, RCOMP_1,
FDIFF_1, CLASSES1, RFINSEQ, MEASURE5, INTEGRA2;
constructors PARTFUN1, REAL_1, SQUARE_1, NAT_1, COMPLEX1, FINSOP_1, RFUNCT_1,
FDIFF_1, RFINSEQ, BINARITH, MEASURE6, FINSEQ_6, INTEGRA2, BINOP_2,
XXREAL_2, NAT_D, RVSUM_1, CLASSES1, SEQ_4, RELSET_1, SEQ_2, COMSEQ_2;
registrations XBOOLE_0, RELAT_1, ORDINAL1, NUMBERS, XXREAL_0, XREAL_0, NAT_1,
MEMBERED, FINSEQ_1, SEQM_3, INTEGRA1, VALUED_0, FUNCT_2, FINSET_1,
XXREAL_2, CARD_1, SEQ_2, MEASURE5, RELSET_1;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
definitions TARSKI;
equalities RVSUM_1;
expansions TARSKI;
theorems SEQ_4, SUBSET_1, PARTFUN1, INTEGRA1, RFUNCT_1, FUNCT_1, FINSEQ_1,
RVSUM_1, SEQ_1, SEQ_2, FDIFF_1, ABSVALUE, NAT_1, RFINSEQ, TARSKI,
FINSEQ_5, FINSEQ_3, FINSEQ_4, FINSEQ_6, INTEGRA2, RELAT_1, XREAL_0,
FUNCT_2, XBOOLE_0, XBOOLE_1, XREAL_1, COMPLEX1, XXREAL_0, FINSOP_1,
ORDINAL1, XXREAL_2, SEQM_3, CLASSES1, MEASURE5;
schemes FINSEQ_2, NAT_1, SEQ_1;
begin :: Lemmas of Division
reserve a,b,e,r,x,y for Real,
i,j,k,n,m for Element of NAT,
x1 for set,
p,q for FinSequence of REAL,
A for non empty closed_interval Subset of REAL,
D,D1,D2 for Division of A,
f,g for Function of A,REAL,
T for DivSequence of A;
Lm1: j-'j+1 = 1
proof
j-'j=j-j by XREAL_1:233
.= 0;
hence thesis;
end;
Lm2: for n st 1 <= n & n <= 2 holds n = 1 or n = 2
proof
let n;
assume that
A1: 1<=n and
A2: n<=2;
per cases by A1,XXREAL_0:1;
suppose
n=1;
hence thesis;
end;
suppose
n>1;
then n >= 1+1 by NAT_1:13;
hence thesis by A2,XXREAL_0:1;
end;
end;
definition
let A be non empty closed_interval Subset of REAL, D be Division of A;
func delta(D) -> Real equals
max rng upper_volume(chi(A,A),D);
correctness;
end;
definition
let A be non empty closed_interval Subset of REAL, T be DivSequence of A;
func delta(T) -> Real_Sequence means :Def2:
for i holds it.i = delta(T.i);
existence
proof
deffunc F(Nat)=delta(T.In($1,NAT));
consider IT being Real_Sequence such that
A1: for i being Nat holds IT.i = F(i)
from SEQ_1:sch 1;
take IT;
let i;
IT.i = F(i) by A1;
hence thesis;
end;
uniqueness
proof
let F1,F2 be Real_Sequence such that
A2: for i holds F1.i = delta(T.i) and
A3: for i holds F2.i = delta(T.i);
for i holds F1.i = F2.i
proof
let i;
F1.i = delta(T.i) by A2
.= F2.i by A3;
hence thesis;
end;
hence thesis by FUNCT_2:63;
end;
end;
theorem
D1 <= D2 implies delta(D1) >= delta(D2)
proof
delta(D2) in rng upper_volume(chi(A,A),D2) by XXREAL_2:def 8;
then consider j such that
A1: j in dom upper_volume(chi(A,A),D2) and
A2: delta(D2) = upper_volume(chi(A,A),D2).j by PARTFUN1:3;
len upper_volume(chi(A,A),D2) = len D2 by INTEGRA1:def 6;
then
A3: j in dom D2 by A1,FINSEQ_3:29;
then
A4: delta(D2) = vol(divset(D2,j)) by A2,INTEGRA1:20;
assume D1 <= D2;
then consider i being Nat such that
A5: i in dom D1 and
A6: divset(D2,j) c= divset(D1,i) by A3,INTEGRA2:37;
A7: vol(divset(D1,i)) = upper_volume(chi(A,A),D1).i by A5,INTEGRA1:20;
len upper_volume(chi(A,A),D1) = len D1 by INTEGRA1:def 6;
then i in dom upper_volume(chi(A,A),D1) by A5,FINSEQ_3:29;
then vol(divset(D1,i)) in rng upper_volume(chi(A,A),D1) by A7,FUNCT_1:def 3;
then delta(D2) <= max rng upper_volume(chi(A,A),D1)
by A4,A6,INTEGRA2:38,XXREAL_2:61;
hence thesis;
end;
theorem Th2:
vol(A) <> 0 implies ex i st i in dom D & vol(divset(D,i)) > 0
proof
assume
A1: vol(A) <>0;
A2: len D in dom D by FINSEQ_5:6;
assume
A3: for i st i in dom D holds vol(divset(D,i))<=0;
A4: i in dom D implies vol(divset(D,i))=0
proof
assume i in dom D;
then vol(divset(D,i)) <= 0 by A3;
hence thesis by INTEGRA1:9;
end;
A5: i in dom D implies upper_bound divset(D,i)=lower_bound divset(D,i)
proof
assume i in dom D;
then vol(divset(D,i))=0 by A4;
then upper_bound divset(D,i)-lower_bound divset(D,i)=0 by INTEGRA1:def 5;
hence thesis;
end;
A6: len D = 1
proof
len D < len D+1 by NAT_1:13;
then
A7: len D-1 < len D by XREAL_1:19;
assume
A8: len D <> 1;
then
A9: upper_bound divset(D,len D)=D.(len D) by A2,INTEGRA1:def 4;
A10: len D-1 in dom D by A2,A8,INTEGRA1:7;
lower_bound divset(D,len D)=D.(len D-1) by A2,A8,INTEGRA1:def 4;
then lower_bound divset(D,len D)<upper_bound divset(D,len D) by A2,A9,A10
,A7,SEQM_3:def 1;
hence contradiction by A5,A2;
end;
then upper_bound divset(D,len D)=D.(len D) by A2,INTEGRA1:def 4;
then
A11: upper_bound divset(D,len D)=upper_bound A by INTEGRA1:def 2;
lower_bound divset(D,len D)=lower_bound A by A2,A6,INTEGRA1:def 4;
then upper_bound A = lower_bound A+0 by A5,A2,A11;
then upper_bound A- lower_bound A = 0;
hence contradiction by A1,INTEGRA1:def 5;
end;
theorem Th3:
x in A implies ex j st j in dom D & x in divset(D,j)
proof
assume
A1: x in A;
then
A2: lower_bound A <= x by INTEGRA2:1;
A3: x <= upper_bound A by A1,INTEGRA2:1;
A4: rng D <> {};
then
A5: 1 in dom D by FINSEQ_3:32;
per cases;
suppose
x in rng D;
then consider j such that
A6: j in dom D and
A7: D.j = x by PARTFUN1:3;
x in divset(D,j)
proof
per cases;
suppose
A8: j=1;
A9: ex a,b st a <= b & a=lower_bound divset(D,j) & b= upper_bound
divset(D,j) by SEQ_4:11;
upper_bound divset(D,j)=D.j by A6,A8,INTEGRA1:def 4;
hence thesis by A7,A9,INTEGRA2:1;
end;
suppose
A10: j<>1;
A11: ex a,b st a <= b & a=lower_bound divset(D,j) & b= upper_bound
divset(D,j) by SEQ_4:11;
upper_bound divset(D,j)=D.j by A6,A10,INTEGRA1:def 4;
hence thesis by A7,A11,INTEGRA2:1;
end;
end;
hence thesis by A6;
end;
suppose
A12: not x in rng D;
defpred MIN[Nat] means x < upper_bound divset(D,$1) & $1 in dom D;
A13: len D in dom D by FINSEQ_5:6;
upper_bound divset(D,len D)=D.(len D)
proof
per cases;
suppose
len D=1;
hence thesis by A13,INTEGRA1:def 4;
end;
suppose
len D<>1;
hence thesis by A13,INTEGRA1:def 4;
end;
end;
then
A14: upper_bound divset(D,len D) = upper_bound A by INTEGRA1:def 2;
x <> upper_bound A
proof
assume x = upper_bound A;
then x = D.(len D) by INTEGRA1:def 2;
hence contradiction by A12,A13,FUNCT_1:def 3;
end;
then x < upper_bound divset(D,len D) by A3,A14,XXREAL_0:1;
then
A15: ex k be Nat st MIN[k] by A13;
consider k be Nat such that
A16: MIN[k] & for n be Nat st MIN[n] holds k <= n from NAT_1:sch 5(A15
);
defpred MAX[Nat] means x >= lower_bound divset(D,$1) & $1 in dom D;
lower_bound divset(D,1)=lower_bound A by A5,INTEGRA1:def 4;
then
A17: ex k be Nat st MAX[k] by A2,A4,FINSEQ_3:32;
A18: for k be Nat holds MAX[k] implies k <= len D by FINSEQ_3:25;
consider j be Nat such that
A19: MAX[j] & for n be Nat st MAX[n] holds n <= j from NAT_1:sch 6(A18
,A17);
k=j
proof
assume
A20: k<>j;
per cases by A20,XXREAL_0:1;
suppose
A21: k < j;
A22: upper_bound divset(D,k)=D.k
proof
per cases;
suppose
k=1;
hence thesis by A16,INTEGRA1:def 4;
end;
suppose
k<>1;
hence thesis by A16,INTEGRA1:def 4;
end;
end;
A23: 1 <= k by A16,FINSEQ_3:25;
then D.(j-1) <= x by A19,A21,INTEGRA1:def 4;
then
A24: D.(j-1) < D.k by A16,A22,XXREAL_0:2;
j-1 in dom D by A19,A21,A23,INTEGRA1:7;
then j-1 < k by A16,A24,SEQ_4:137;
then j < k+1 by XREAL_1:19;
hence contradiction by A21,NAT_1:13;
end;
suppose
A25: k > j;
x < upper_bound divset(D,j)
proof
A26: upper_bound divset(D,j)=D.j
proof
per cases;
suppose
j=1;
hence thesis by A19,INTEGRA1:def 4;
end;
suppose
j<>1;
hence thesis by A19,INTEGRA1:def 4;
end;
end;
assume
A27: x >= upper_bound divset(D,j);
A28: j+1 <= k by A25,NAT_1:13;
A29: 1 <= j by A19,FINSEQ_3:25;
then
A30: 1 <= j+1 by NAT_1:13;
k <= len D by A16,FINSEQ_3:25;
then j+1 <= len D by A28,XXREAL_0:2;
then
A31: j+1 in dom D by A30,FINSEQ_3:25;
j+1 > 1 by A29,NAT_1:13;
then lower_bound divset(D,j+1) = D.(j+1-1) by A31,INTEGRA1:def 4
.=D.j;
then j+1 <= j by A19,A27,A26,A31;
hence contradiction by NAT_1:13;
end;
hence contradiction by A16,A19,A25;
end;
end;
then x in divset(D,k) by A16,A19,INTEGRA2:1;
hence thesis by A16;
end;
end;
theorem Th4:
ex D st D1 <= D & D2 <= D & rng D = rng D1 \/ rng D2
proof
consider D being FinSequence of REAL such that
A1: rng D = rng(D1^D2) and
A2: len D = card rng(D1^D2) and
A3: D is increasing by SEQ_4:140;
reconsider D as increasing FinSequence of REAL by A3;
reconsider D as non empty increasing FinSequence of REAL by A1;
A4: rng D2 c= A by INTEGRA1:def 2;
A5: rng(D1^D2) = rng D1 \/ rng D2 by FINSEQ_1:31;
then
A6: rng D1 c= rng(D1^D2) by XBOOLE_1:7;
rng D1 c= A by INTEGRA1:def 2;
then
A7: rng D c= A by A1,A5,A4,XBOOLE_1:8;
D.len D = upper_bound A
proof
len D1 in dom D1 by FINSEQ_5:6;
then D1.len D1 in rng D1 by FUNCT_1:def 3;
then consider k such that
A8: k in dom D and
A9: D1.len D1=D.k by A1,A6,PARTFUN1:3;
assume
A10: D.len D <> upper_bound A;
A11: len D in dom D by FINSEQ_5:6;
then D.len D in rng D by FUNCT_1:def 3;
then D.len D <= upper_bound A by A7,INTEGRA2:1;
then
A12: D.len D < upper_bound A by A10,XXREAL_0:1;
D1.len D1 = upper_bound A by INTEGRA1:def 2;
then k > len D by A11,A12,A8,A9,SEQ_4:137;
hence contradiction by A8,FINSEQ_3:25;
end;
then reconsider D as Division of A by A7,INTEGRA1:def 2;
take D;
card rng D1 <= len D by A2,A5,NAT_1:43,XBOOLE_1:7;
then len D1 <= len D by FINSEQ_4:62;
hence D1 <= D by A1,A6,INTEGRA1:def 18;
A13: rng D2 c= rng(D1^D2) by A5,XBOOLE_1:7;
card rng D2 <= len D by A2,A5,NAT_1:43,XBOOLE_1:7;
then len D2 <= len D by FINSEQ_4:62;
hence D2 <= D by A1,A13,INTEGRA1:def 18;
thus thesis by A1,FINSEQ_1:31;
end;
theorem Th5:
delta(D1) < min rng upper_volume(chi(A,A),D) implies for x,y,i st
i in dom D1 & x in rng D /\ divset(D1,i) & y in rng D /\ divset(D1,i) holds x=y
proof
assume
A1: delta(D1)<min rng upper_volume(chi(A,A),D);
let x,y,i;
assume
A2: i in dom D1;
assume
A3: x in rng D /\ divset(D1,i);
then x in rng D by XBOOLE_0:def 4;
then consider n such that
A4: n in dom D and
A5: x=D.n by PARTFUN1:3;
assume
A6: y in rng D /\ divset(D1,i);
then y in rng D by XBOOLE_0:def 4;
then consider m such that
A7: m in dom D and
A8: y=D.m by PARTFUN1:3;
assume
A9: x<>y;
A10: |.D.n-D.m.| >= min rng upper_volume(chi(A,A),D)
proof
per cases by A9,A5,A8,XXREAL_0:1;
suppose
n<m;
then
A11: n+1<=m by NAT_1:13;
A12: 1<=n+1 by NAT_1:12;
m in Seg len D by A7,FINSEQ_1:def 3;
then m<=len D by FINSEQ_1:1;
then n+1<=len D by A11,XXREAL_0:2;
then
A13: n+1 in Seg len D by A12,FINSEQ_1:1;
then
A14: n+1 in dom D by FINSEQ_1:def 3;
then D.m>=D.(n+1) by A7,A11,SEQ_4:137;
then D.n-D.m <= D.n-D.(n+1) by XREAL_1:10;
then
A15: -(D.n-D.m) >= -(D.n-D.(n+1)) by XREAL_1:24;
n+1 in Seg len upper_volume(chi(A,A),D) by A13,INTEGRA1:def 6;
then n+1 in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3;
then
A16: (upper_volume(chi(A,A),D)).(n+1) in rng upper_volume(chi(A,A),D) by
FUNCT_1:def 3;
n in Seg len D by A4,FINSEQ_1:def 3;
then 1<=n by FINSEQ_1:1;
then
A17: n+1 <> 1 by NAT_1:13;
then
A18: upper_bound divset(D,n+1)=D.(n+1) by A14,INTEGRA1:def 4;
-|.D.n-D.m.| <= D.n-D.m by ABSVALUE:4;
then
A19: |.D.n-D.m.| >= -(D.n-D.m) by XREAL_1:26;
lower_bound divset(D,n+1)=D.((n+1)-1) by A14,A17,INTEGRA1:def 4;
then vol(divset(D,n+1))=D.(n+1)-D.n by A18,INTEGRA1:def 5;
then D.(n+1)-D.n=(upper_volume(chi(A,A),D)).(n+1) by A14,INTEGRA1:20;
then D.(n+1)-D.n>=min rng upper_volume(chi(A,A),D) by A16,XXREAL_2:def 7;
then -(D.n-D.m) >= min rng upper_volume(chi(A,A),D) by A15,XXREAL_0:2;
hence thesis by A19,XXREAL_0:2;
end;
suppose
n>m;
then
A20: m+1<=n by NAT_1:13;
n in Seg len D by A4,FINSEQ_1:def 3;
then n<=len D by FINSEQ_1:1;
then
A21: m+1<=len D by A20,XXREAL_0:2;
A22: 1<=m+1 by NAT_1:12;
then
A23: m+1 in dom D by A21,FINSEQ_3:25;
then D.(m+1)<=D.n by A4,A20,SEQ_4:137;
then
A24: D.n-D.m >= D.(m+1)-D.m by XREAL_1:9;
m+1 in Seg len D by A22,A21,FINSEQ_1:1;
then m+1 in Seg len upper_volume(chi(A,A),D) by INTEGRA1:def 6;
then m+1 in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3;
then
A25: (upper_volume(chi(A,A),D)).(m+1) in rng upper_volume(chi(A,A),D) by
FUNCT_1:def 3;
m in Seg len D by A7,FINSEQ_1:def 3;
then 1<=m by FINSEQ_1:1;
then
A26: 1 < m+1 by NAT_1:13;
then
A27: upper_bound divset(D,m+1)=D.(m+1) by A23,INTEGRA1:def 4;
lower_bound divset(D,m+1)=D.((m+1)-1) by A23,A26,INTEGRA1:def 4;
then vol(divset(D,m+1))=D.(m+1)-D.m by A27,INTEGRA1:def 5;
then D.(m+1)-D.m=(upper_volume(chi(A,A),D)).(m+1) by A23,INTEGRA1:20;
then D.(m+1)-D.m>=min rng upper_volume(chi(A,A),D) by A25,XXREAL_2:def 7;
then
A28: D.n-D.m >= min rng upper_volume(chi(A,A),D) by A24,XXREAL_0:2;
|.D.n-D.m.| >= D.n-D.m by ABSVALUE:4;
hence thesis by A28,XXREAL_0:2;
end;
end;
|.D.n-D.m.| <= delta D1
proof
per cases by A9,A5,A8,XXREAL_0:1;
suppose
A29: n<m;
i in Seg len D1 by A2,FINSEQ_1:def 3;
then i in Seg len upper_volume(chi(A,A),D1) by INTEGRA1:def 6;
then i in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3;
then upper_volume(chi(A,A),D1).i in rng upper_volume(chi(A,A),D1) by
FUNCT_1:def 3;
then
upper_volume(chi(A,A),D1).i <= max rng upper_volume(chi(A,A),D1) by
XXREAL_2:def 8;
then
A30: upper_volume(chi(A,A),D1).i<=delta(D1);
D.m in divset(D1,i) by A6,A8,XBOOLE_0:def 4;
then D.m <= upper_bound divset(D1,i) by INTEGRA2:1;
then
A31: D.m-lower_bound divset(D1,i)<= upper_bound divset(D1,i)-lower_bound
divset(D1,i) by XREAL_1:9;
D.n in divset(D1,i) by A3,A5,XBOOLE_0:def 4;
then D.n >= lower_bound divset(D1,i) by INTEGRA2:1;
then D.m-D.n <= D.m-lower_bound divset(D1,i) by XREAL_1:10;
then D.m-D.n <= upper_bound divset(D1,i)- lower_bound divset(D1,i) by A31
,XXREAL_0:2;
then D.m-D.n <= vol(divset(D1,i)) by INTEGRA1:def 5;
then
A32: D.m-D.n <= upper_volume(chi(A,A),D1).i by A2,INTEGRA1:20;
D.n<D.m by A4,A7,A29,SEQM_3:def 1;
then D.n-D.m<0 by XREAL_1:49;
then |.D.n-D.m.|=-(D.n-D.m) by ABSVALUE:def 1
.= D.m-D.n;
hence thesis by A32,A30,XXREAL_0:2;
end;
suppose
A33: n>m;
i in Seg len D1 by A2,FINSEQ_1:def 3;
then i in Seg len upper_volume(chi(A,A),D1) by INTEGRA1:def 6;
then i in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3;
then upper_volume(chi(A,A),D1).i in rng upper_volume(chi(A,A),D1) by
FUNCT_1:def 3;
then
upper_volume(chi(A,A),D1).i <= max rng upper_volume(chi(A,A),D1) by
XXREAL_2:def 8;
then
A34: upper_volume(chi(A,A),D1).i<=delta(D1);
D.n in divset(D1,i) by A3,A5,XBOOLE_0:def 4;
then D.n <= upper_bound divset(D1,i) by INTEGRA2:1;
then
A35: D.n-lower_bound divset(D1,i)<= upper_bound divset(D1,i)-
lower_bound divset(D1,i) by XREAL_1:9;
D.m in divset(D1,i) by A6,A8,XBOOLE_0:def 4;
then D.m >= lower_bound divset(D1,i) by INTEGRA2:1;
then D.n-D.m <= D.n-lower_bound divset(D1,i) by XREAL_1:10;
then
D.n-D.m <= upper_bound divset(D1,i)-lower_bound divset(D1,i) by A35,
XXREAL_0:2;
then D.n-D.m <= vol(divset(D1,i)) by INTEGRA1:def 5;
then
A36: D.n-D.m <= upper_volume(chi(A,A),D1).i by A2,INTEGRA1:20;
D.n>D.m by A4,A7,A33,SEQM_3:def 1;
then D.n-D.m>0 by XREAL_1:50;
then |.D.n-D.m.|=D.n-D.m by ABSVALUE:def 1;
hence thesis by A36,A34,XXREAL_0:2;
end;
end;
hence contradiction by A1,A10,XXREAL_0:2;
end;
theorem Th6:
for p,q st rng p = rng q & p is increasing & q is increasing
holds p = q
proof
let p,q;
assume
A1: rng p = rng q;
assume that
A2: p is increasing and
A3: q is increasing;
A4: q is one-to-one by A3;
p is one-to-one by A2;
then len p = len q by A1,A4,FINSEQ_1:48;
hence thesis by A1,A2,A3,SEQ_4:141;
end;
theorem Th7:
D <= D1 & i in dom D & j in dom D & i <= j implies indx(D1,D,i)
<= indx(D1,D,j) & indx(D1,D,i) in dom D1
proof
assume that
A1: D <= D1 and
A2: i in dom D and
A3: j in dom D and
A4: i <= j;
A5: D.i = D1.indx(D1,D,i) by A1,A2,INTEGRA1:def 19;
A6: indx(D1,D,j) in dom D1 by A1,A3,INTEGRA1:def 19;
A7: D.j = D1.indx(D1,D,j) by A1,A3,INTEGRA1:def 19;
A8: indx(D1,D,i) in dom D1 by A1,A2,INTEGRA1:def 19;
D.i <= D.j by A2,A3,A4,SEQ_4:137;
hence thesis by A5,A8,A7,A6,SEQM_3:def 1;
end;
theorem Th8:
D <= D1 & i in dom D & j in dom D & i < j implies indx(D1,D,i) <
indx(D1,D,j)
proof
assume that
A1: D <= D1 and
A2: i in dom D and
A3: j in dom D and
A4: i < j;
A5: D.i = D1.indx(D1,D,i) by A1,A2,INTEGRA1:def 19;
A6: indx(D1,D,j) in dom D1 by A1,A3,INTEGRA1:def 19;
A7: D.j = D1.indx(D1,D,j) by A1,A3,INTEGRA1:def 19;
A8: indx(D1,D,i) in dom D1 by A1,A2,INTEGRA1:def 19;
D.i < D.j by A2,A3,A4,SEQM_3:def 1;
hence thesis by A5,A8,A7,A6,SEQ_4:137;
end;
theorem Th9:
delta(D) >= 0
proof
consider y being Element of REAL such that
A1: y in rng D by SUBSET_1:4;
consider n such that
A2: n in dom D and
y=D.n by A1,PARTFUN1:3;
n in Seg len D by A2,FINSEQ_1:def 3;
then n in Seg len upper_volume(chi(A,A),D) by INTEGRA1:def 6;
then n in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3;
then upper_volume(chi(A,A),D).n in rng upper_volume(chi(A,A),D) by
FUNCT_1:def 3;
then
A3: upper_volume(chi(A,A),D).n <= max rng upper_volume(chi(A,A),D) by
XXREAL_2:def 8;
vol(divset(D,n))=upper_volume(chi(A,A),D).n by A2,INTEGRA1:20;
then upper_volume(chi(A,A),D).n >= 0 by INTEGRA1:9;
hence thesis by A3;
end;
Lm3: for A be non empty closed_interval Subset of REAL,
g be Function of A,REAL st g|A
is bounded holds upper_bound rng g >= lower_bound rng g
proof
let A be non empty closed_interval Subset of REAL;
let g be Function of A,REAL;
assume
A1: g|A is bounded;
then
A2: rng g is bounded_below by INTEGRA1:11;
rng g is bounded_above by A1,INTEGRA1:13;
hence thesis by A2,SEQ_4:11;
end;
Lm4: for A,B be non empty closed_interval Subset of REAL,
f be Function of A,REAL st f|A
is bounded & B c= A holds lower_bound rng(f|B) >= lower_bound rng f &
lower_bound rng f <= upper_bound rng(f|B) & upper_bound rng(f|B) <= upper_bound
rng f & lower_bound rng(f|B) <= upper_bound rng f
proof
let A,B be non empty closed_interval Subset of REAL, f be Function of A,REAL;
assume that
A1: f|A is bounded and
A2: B c= A;
B c=dom f by A2,FUNCT_2:def 1;
then
A3: dom(f|B)=B by RELAT_1:62;
then
A4: rng(f|B) <> {} by RELAT_1:42;
consider x being Element of REAL such that
A5: x in B by SUBSET_1:4;
A6: (f|B).x in rng(f|B) by A5,A3,FUNCT_1:def 3;
A7: rng f is bounded_below by A1,INTEGRA1:11;
hence
A8: lower_bound rng(f|B) >= lower_bound rng f by A4,RELAT_1:70,SEQ_4:47;
rng(f|B) is bounded_below by A7,RELAT_1:70,XXREAL_2:44;
then
A9: lower_bound rng(f|B)<=(f|B).x by A6,SEQ_4:def 2;
A10: rng f is bounded_above by A1,INTEGRA1:13;
then rng(f|B) is bounded_above by RELAT_1:70,XXREAL_2:43;
then upper_bound rng(f|B)>=(f|B).x by A6,SEQ_4:def 1;
then
A11: lower_bound rng(f|B) <= upper_bound rng(f|B) by A9,XXREAL_0:2;
hence upper_bound rng(f|B) >= lower_bound rng f by A8,XXREAL_0:2;
thus upper_bound rng(f|B) <= upper_bound rng f by A10,A4,RELAT_1:70,SEQ_4:48;
hence thesis by A11,XXREAL_0:2;
end;
Lm5: j in dom D1 implies vol(divset(D1,j)) <= delta D1
proof
assume
A1: j in dom D1;
then j in Seg len D1 by FINSEQ_1:def 3;
then j in Seg len upper_volume(chi(A,A),D1) by INTEGRA1:def 6;
then j in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3;
then upper_volume(chi(A,A),D1).j in rng upper_volume(chi(A,A),D1) by
FUNCT_1:def 3;
then upper_volume(chi(A,A),D1).j <= max rng upper_volume(chi(A,A),D1) by
XXREAL_2:def 8;
then vol(divset(D1,j)) <= max rng upper_volume(chi(A,A),D1) by A1,INTEGRA1:20
;
hence thesis;
end;
Lm6: for j1 be Element of NAT st j1 = len D1 - 1 & x in divset(D1,len D1) &
len D1 >= 2 & D1 <= D2 & rng D2 = rng D1 \/ {x} holds rng (D2|indx(D2,D1,j1)) =
rng (D1|j1)
proof
let j1 be Element of NAT;
assume that
A1: j1 = len D1 - 1 and
A2: x in divset(D1,len D1) and
A3: len D1 >= 2;
A4: len D1 in dom D1 by FINSEQ_5:6;
assume that
A5: D1<=D2 and
A6: rng D2 = rng D1 \/ {x};
A7: len D1 <> 1 by A3;
then
A8: len D1-1 in dom D1 by A4,INTEGRA1:7;
then
A9: indx(D2,D1,j1) in dom D2 by A1,A5,INTEGRA1:def 19;
then
A10: indx(D2,D1,j1) <= len D2 by FINSEQ_3:25;
A11: j1 in dom D1 by A1,A4,A7,INTEGRA1:7;
then
A12: 1 <= j1 by FINSEQ_3:25;
A13: j1 <= len D1 by A11,FINSEQ_3:25;
lower_bound divset(D1,len D1) <= x by A2,INTEGRA2:1;
then
A14: D1.j1 <= x by A1,A4,A7,INTEGRA1:def 4;
for x1 being object st x1 in rng(D2|indx(D2,D1,j1)) holds x1 in rng(D1|j1)
proof
let x1 be object;
assume x1 in rng(D2|indx(D2,D1,j1));
then consider k such that
A15: k in dom(D2|indx(D2,D1,j1)) and
A16: x1=(D2|indx(D2,D1,j1)).k by PARTFUN1:3;
k in Seg len(D2|indx(D2,D1,j1)) by A15,FINSEQ_1:def 3;
then
A17: k in Seg indx(D2,D1,j1) by A10,FINSEQ_1:59;
then
A18: k in dom D2 by A9,RFINSEQ:6;
A19: len(D1|j1) = j1 by A13,FINSEQ_1:59;
k <= indx(D2,D1,j1) by A17,FINSEQ_1:1;
then D2.k <= D2.indx(D2,D1,j1) by A9,A18,SEQ_4:137;
then
A20: D2.k <= D1.j1 by A1,A5,A8,INTEGRA1:def 19;
A21: D2.k in rng D1 implies D2.k in rng(D1|j1)
proof
assume D2.k in rng D1;
then consider m such that
A22: m in dom D1 and
A23: D2.k = D1.m by PARTFUN1:3;
m in Seg len D1 by A22,FINSEQ_1:def 3;
then
A24: 1 <= m by FINSEQ_1:1;
A25: m <= j1 by A11,A20,A22,A23,SEQM_3:def 1;
then m in Seg j1 by A24,FINSEQ_1:1;
then
A26: D2.k = (D1|j1).m by A11,A23,RFINSEQ:6;
m in dom (D1|j1) by A19,A24,A25,FINSEQ_3:25;
hence thesis by A26,FUNCT_1:def 3;
end;
A27: D2.k in {x} implies D2.k = D1.j1
proof
assume D2.k in {x};
then D1.j1 <= D2.k by A14,TARSKI:def 1;
hence thesis by A20,XXREAL_0:1;
end;
A28: D2.k in {x} implies D2.k in rng(D1|j1)
proof
j1 in dom(D1|j1) by A12,A19,FINSEQ_3:25;
then
A29: (D1|j1).j1 in rng(D1|j1) by FUNCT_1:def 3;
assume
A30: D2.k in {x};
j1 in Seg j1 by A12,FINSEQ_1:1;
hence thesis by A11,A27,A30,A29,RFINSEQ:6;
end;
D2.k in rng D2 by A18,FUNCT_1:def 3;
hence thesis by A6,A9,A16,A17,A28,A21,RFINSEQ:6,XBOOLE_0:def 3;
end;
then
A31: rng(D2|indx(D2,D1,j1)) c= rng (D1|j1);
for x1 being object st x1 in rng(D1|j1) holds x1 in rng(D2|indx(D2,D1,j1))
proof
let x1 be object;
assume x1 in rng(D1|j1);
then consider k such that
A32: k in dom(D1|j1) and
A33: x1=(D1|j1).k by PARTFUN1:3;
k in Seg len(D1|j1) by A32,FINSEQ_1:def 3;
then
A34: k in Seg j1 by A13,FINSEQ_1:59;
then
A35: k in dom D1 by A11,RFINSEQ:6;
k <= j1 by A34,FINSEQ_1:1;
then D1.k <= D1.j1 by A1,A8,A35,SEQ_4:137;
then D2.indx(D2,D1,k) <= D1.j1 by A5,A35,INTEGRA1:def 19;
then
A36: D2.indx(D2,D1,k)<=D2.indx(D2,D1,j1) by A1,A5,A8,INTEGRA1:def 19;
A37: (D1|j1).k = D1.k by A11,A34,RFINSEQ:6;
D1.k in rng D1 by A35,FUNCT_1:def 3;
then x1 in rng D2 by A6,A33,A37,XBOOLE_0:def 3;
then consider n such that
A38: n in dom D2 and
A39: x1=D2.n by PARTFUN1:3;
D2.indx(D2,D1,k)=D2.n by A5,A33,A37,A35,A39,INTEGRA1:def 19;
then
A40: n <= indx(D2,D1,j1) by A9,A38,A36,SEQM_3:def 1;
1 <= n by A38,FINSEQ_3:25;
then
A41: n in Seg indx(D2,D1,j1) by A40,FINSEQ_1:1;
then n in Seg len(D2|indx(D2,D1,j1)) by A10,FINSEQ_1:59;
then
A42: n in dom(D2|indx(D2,D1,j1)) by FINSEQ_1:def 3;
D2.n = (D2|indx(D2,D1,j1)).n by A9,A41,RFINSEQ:6;
hence thesis by A39,A42,FUNCT_1:def 3;
end;
then rng(D1|j1) c= rng(D2|indx(D2,D1,j1));
hence thesis by A31,XBOOLE_0:def 10;
end;
theorem Th10:
x in divset(D1,len D1) & len D1 >= 2 & D1<=D2 & rng D2 = rng D1
\/ {x} & g|A is bounded implies Sum lower_volume(g,D2) - Sum lower_volume(g,D1)
<= (upper_bound rng g-lower_bound rng g)*delta(D1)
proof
assume that
A1: x in divset(D1,len D1) and
A2: len D1 >= 2;
set j = len D1;
assume that
A3: D1<=D2 and
A4: rng D2 = rng D1 \/ {x};
A5: len D1 in dom D1 by FINSEQ_5:6;
then
A6: indx(D2,D1,j) in dom D2 by A3,INTEGRA1:def 19;
A7: len D1 <> 1 by A2;
then reconsider j1=len D1-1 as Element of NAT by A5,INTEGRA1:7;
A8: j1 in dom D1 by A5,A7,INTEGRA1:7;
then
A9: j1 <= len D1 by FINSEQ_3:25;
A10: 1 <= j1 by A8,FINSEQ_3:25;
then mid(D1,1,j1) is increasing by A5,A7,INTEGRA1:7,35;
then
A11: D1|j1 is increasing by A10,FINSEQ_6:116;
A12: len D1-1 in dom D1 by A5,A7,INTEGRA1:7;
then
A13: indx(D2,D1,j1) in dom D2 by A3,INTEGRA1:def 19;
then
A14: 1 <= indx(D2,D1,j1) by FINSEQ_3:25;
then mid(D2,1,indx(D2,D1,j1)) is increasing by A13,INTEGRA1:35;
then
A15: D2|indx(D2,D1,j1) is increasing by A14,FINSEQ_6:116;
A16: indx(D2,D1,j1) <= len D2 by A13,FINSEQ_3:25;
then
A17: len (D2|indx(D2,D1,j1))=indx(D2,D1,j1) by FINSEQ_1:59;
A18: rng (D2|indx(D2,D1,j1)) = rng (D1|j1) by A1,A2,A3,A4,Lm6;
then
A19: D2|indx(D2,D1,j1)=D1|j1 by A15,A11,Th6;
A20: for k st 1 <= k & k <= j1 holds k=indx(D2,D1,k)
proof
let k;
assume that
A21: 1 <= k and
A22: k <= j1;
assume
A23: k<>indx(D2,D1,k);
per cases by A23,XXREAL_0:1;
suppose
A24: k > indx(D2,D1,k);
k <= len D1 by A9,A22,XXREAL_0:2;
then
A25: k in dom D1 by A21,FINSEQ_3:25;
then indx(D2,D1,k) in dom D2 by A3,INTEGRA1:def 19;
then indx(D2,D1,k) in Seg len D2 by FINSEQ_1:def 3;
then
A26: 1<=indx(D2,D1,k) by FINSEQ_1:1;
A27: indx(D2,D1,k) < j1 by A22,A24,XXREAL_0:2;
then
A28: indx(D2,D1,k) in Seg j1 by A26,FINSEQ_1:1;
indx(D2,D1,k)<=indx(D2,D1,j1) by A3,A8,A22,A25,Th7;
then indx(D2,D1,k) in Seg indx(D2,D1,j1) by A26,FINSEQ_1:1;
then
A29: (D2|indx(D2,D1,j1)).indx(D2,D1,k)=D2.indx(D2,D1,k) by A13,RFINSEQ:6;
indx(D2,D1,k) <= len D1 by A9,A27,XXREAL_0:2;
then indx(D2,D1,k) in dom D1 by A26,FINSEQ_3:25;
then
A30: D1.k > D1.indx(D2,D1,k) by A24,A25,SEQM_3:def 1;
D1.k=D2.indx(D2,D1,k) by A3,A25,INTEGRA1:def 19;
hence contradiction by A8,A19,A29,A30,A28,RFINSEQ:6;
end;
suppose
A31: k < indx(D2,D1,k);
k <= len D1 by A9,A22,XXREAL_0:2;
then
A32: k in dom D1 by A21,FINSEQ_3:25;
then indx(D2,D1,k) <= indx(D2,D1,j1) by A3,A8,A22,Th7;
then
A33: k <= indx(D2,D1,j1) by A31,XXREAL_0:2;
then k <= len D2 by A16,XXREAL_0:2;
then
A34: k in dom D2 by A21,FINSEQ_3:25;
k in Seg j1 by A21,A22,FINSEQ_1:1;
then
A35: D1.k = (D1|j1).k by A12,RFINSEQ:6;
indx(D2,D1,k) in dom D2 by A3,A32,INTEGRA1:def 19;
then
A36: D2.k < D2.indx(D2,D1,k) by A31,A34,SEQM_3:def 1;
A37: k in Seg indx(D2,D1,j1) by A21,A33,FINSEQ_1:1;
D1.k=D2.indx(D2,D1,k) by A3,A32,INTEGRA1:def 19;
hence contradiction by A13,A19,A35,A36,A37,RFINSEQ:6;
end;
end;
A38: for k be Nat st 1 <= k & k <= len(lower_volume(g,D1)|j1) holds (
lower_volume(g,D1)|j1).k = (lower_volume(g,D2)|indx(D2,D1,j1)).k
proof
indx(D2,D1,j1) in Seg len D2 by A13,FINSEQ_1:def 3;
then indx(D2,D1,j1) in Seg len lower_volume(g,D2) by INTEGRA1:def 7;
then
A39: indx(D2,D1,j1) in dom lower_volume(g,D2) by FINSEQ_1:def 3;
let k be Nat;
assume that
A40: 1 <= k and
A41: k <= len(lower_volume(g,D1)|j1);
reconsider k as Element of NAT by ORDINAL1:def 12;
A42: len(lower_volume(g,D1)) = len D1 by INTEGRA1:def 7;
then
A43: k <= j1 by A9,A41,FINSEQ_1:59;
then k <= len D1 by A9,XXREAL_0:2;
then k in Seg len D1 by A40,FINSEQ_1:1;
then
A44: k in dom D1 by FINSEQ_1:def 3;
then
A45: indx(D2,D1,k) in dom D2 by A3,INTEGRA1:def 19;
A46: k in Seg j1 by A40,A43,FINSEQ_1:1;
then indx(D2,D1,k) in Seg j1 by A20,A40,A43;
then
A47: indx(D2,D1,k) in Seg indx(D2,D1,j1) by A10,A20;
then indx(D2,D1,k)<=indx(D2,D1,j1) by FINSEQ_1:1;
then
A48: indx(D2,D1,k)<=len D2 by A16,XXREAL_0:2;
A49: D1.k=D2.indx(D2,D1,k) by A3,A44,INTEGRA1:def 19;
A50: lower_bound divset(D1,k)=lower_bound divset(D2,indx(D2,D1,k)) &
upper_bound divset(D1,k)=upper_bound divset(D2,indx(D2,D1,k))
proof
per cases;
suppose
A51: k=1;
then
A52: upper_bound divset(D1,k)=D1.k by A44,INTEGRA1:def 4;
A53: lower_bound divset(D1,k)=lower_bound A by A44,A51,INTEGRA1:def 4;
indx(D2,D1,k)=1 by A10,A20,A51;
hence thesis by A45,A49,A53,A52,INTEGRA1:def 4;
end;
suppose
A54: k<>1;
then reconsider k1=k-1 as Element of NAT by A44,INTEGRA1:7;
A55: k-1 in dom D1 by A44,A54,INTEGRA1:7;
then
A56: 1 <= k1 by FINSEQ_3:25;
k <= k+1 by NAT_1:11;
then k1 <= k by XREAL_1:20;
then
A57: k1 <= j1 by A43,XXREAL_0:2;
A58: indx(D2,D1,k)<>1 by A20,A40,A43,A54;
then
A59: lower_bound divset(D2,indx(D2,D1,k))=D2.(indx(D2,D1,k)-1) by A45,
INTEGRA1:def 4;
A60: upper_bound divset(D1,k)=D1.k by A44,A54,INTEGRA1:def 4;
A61: lower_bound divset(D1,k)=D1.(k-1) by A44,A54,INTEGRA1:def 4;
A62: upper_bound divset(D2,indx(D2,D1,k))=D2.indx(D2,D1,k) by A45,A58,
INTEGRA1:def 4;
D2.(indx(D2,D1,k)-1)=D2.(k-1) by A20,A40,A43
.= D2.indx(D2,D1,k1) by A20,A57,A56;
hence thesis by A3,A44,A61,A60,A55,A59,A62,INTEGRA1:def 19;
end;
end;
divset(D1,k)=[. lower_bound divset(D1,k), upper_bound divset(D1,k) .]
by INTEGRA1:4;
then
A63: divset(D1,k)=divset(D2,indx(D2,D1,k)) by A50,INTEGRA1:4;
j1 in Seg len(lower_volume(g,D1)) by A8,A42,FINSEQ_1:def 3;
then j1 in dom(lower_volume(g,D1)) by FINSEQ_1:def 3;
then
A64: (lower_volume(g,D1)|j1).k = lower_volume(g,D1).k by A46,RFINSEQ:6
.=(lower_bound(rng(g|divset(D2,indx(D2,D1,k)))))* vol(divset(D2,indx(
D2,D1,k))) by A44,A63,INTEGRA1:def 7;
1<=indx(D2,D1,k) by A20,A40,A43;
then
A65: indx(D2,D1,k) in dom D2 by A48,FINSEQ_3:25;
(lower_volume(g,D2)|indx(D2,D1,j1)).k =(lower_volume(g,D2)|indx(D2,
D1,j1)).indx(D2,D1,k) by A20,A40,A43
.=lower_volume(g,D2).indx(D2,D1,k) by A47,A39,RFINSEQ:6
.=(lower_bound(rng(g|divset(D2,indx(D2,D1,k)))))* vol(divset(D2,indx(
D2,D1,k))) by A65,INTEGRA1:def 7;
hence thesis by A64;
end;
A66: len D2 in dom D2 by FINSEQ_5:6;
deffunc LVg(Division of A) = lower_volume(g,$1);
deffunc PLg(Division of A,Nat) = (PartSums(lower_volume(g,$1))).$2;
A67: j >= len lower_volume(g,D1) by INTEGRA1:def 7;
A68: j <= len LVg(D1) by INTEGRA1:def 7;
A69: len D1 in Seg len D1 by FINSEQ_1:3;
then
A70: 1 <= j by FINSEQ_1:1;
then
A71: j in dom LVg(D1) by A68,FINSEQ_3:25;
assume
A72: g|A is bounded;
j < j+1 by NAT_1:13;
then
A73: j1 < j by XREAL_1:19;
then j1 < len LVg(D1) by INTEGRA1:def 7;
then j1 in dom LVg(D1) by A10,FINSEQ_3:25;
then PLg(D1,j1)=Sum(LVg(D1)|j1) by INTEGRA1:def 20;
then PLg(D1,j1)+Sum mid(LVg(D1),j,j) =Sum((LVg(D1)|j1)^mid(LVg(D1),j,j)) by
RVSUM_1:75
.=Sum(mid(LVg(D1),1,j1)^mid(LVg(D1),j1+1,j)) by A10,FINSEQ_6:116
.=Sum(mid(LVg(D1),1,j)) by A10,A68,A73,INTEGRA2:4
.=Sum(LVg(D1)|j) by A70,FINSEQ_6:116;
then
A74: PLg(D1,j1)+Sum mid(lower_volume(g,D1),j,j)=PLg(D1,j) by A71,
INTEGRA1:def 20;
A75: indx(D2,D1,j) in dom D2 by A3,A5,INTEGRA1:def 19;
then
A76: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3;
then
A77: 1 <= indx(D2,D1,j) by FINSEQ_1:1;
len D1 < len D1+1 by NAT_1:13;
then j1 < len D1 by XREAL_1:19;
then
A78: indx(D2,D1,j1) < indx(D2,D1,len D1) by A3,A5,A8,Th8;
then
A79: indx(D2,D1,j1)+1 <= indx(D2,D1,len D1) by NAT_1:13;
A80: j1 in dom D1 by A5,A7,INTEGRA1:7;
A81: Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,len D1)) -Sum
mid(lower_volume(g,D1),len D1,len D1) <= (upper_bound rng g-lower_bound rng g)*
delta(D1)
proof
A82: indx(D2,D1,j)-indx(D2,D1,j1) <= 2
proof
set ID1=indx(D2,D1,j1)+1;
set ID2=ID1+1;
assume indx(D2,D1,j)-indx(D2,D1,j1) > 2;
then
A83: indx(D2,D1,j1)+(1+1)<indx(D2,D1,j) by XREAL_1:20;
A84: ID1 < ID2 by NAT_1:13;
then indx(D2,D1,j1) <= ID2 by NAT_1:13;
then
A85: 1 <= ID2 by A14,XXREAL_0:2;
A86: indx(D2,D1,j) in dom D2 by A3,A5,INTEGRA1:def 19;
then
A87: indx(D2,D1,j) <= len D2 by FINSEQ_3:25;
then ID2 <= len D2 by A83,XXREAL_0:2;
then
A88: ID2 in dom D2 by A85,FINSEQ_3:25;
then
A89: D2.ID2<D2.indx(D2, D1,j) by A83,A86,SEQM_3:def 1;
A90: 1 <= ID1 by A14,NAT_1:13;
A91: D1.j1 = D2.indx(D2,D1,j1) by A3,A8,INTEGRA1:def 19;
ID1 <= indx(D2,D1,j) by A83,A84,XXREAL_0:2;
then ID1 <= len D2 by A87,XXREAL_0:2;
then
A92: ID1 in dom D2 by A90,FINSEQ_3:25;
A93: D1.j = D2.indx(D2,D1,j) by A3,A5,INTEGRA1:def 19;
indx(D2,D1,j1) < ID1 by NAT_1:13;
then
A94: D2.indx(D2,D1,j1)<D2.ID1 by A13,A92,SEQM_3:def 1;
A95: D2.ID1<D2.ID2 by A84,A92,A88,SEQM_3:def 1;
A96: not D2.ID1 in rng D1 & not D2.ID2 in rng D1
proof
assume
A97: D2.ID1 in rng D1 or D2.ID2 in rng D1;
per cases by A97;
suppose
D2.ID1 in rng D1;
then consider n such that
A98: n in dom D1 and
A99: D1.n=D2.ID1 by PARTFUN1:3;
j1<n by A80,A94,A91,A98,A99,SEQ_4:137;
then
A100: j<n+1 by XREAL_1:19;
D2.ID1<D2.indx(D2,D1,j) by A95,A89,XXREAL_0:2;
then n<j by A5,A93,A98,A99,SEQ_4:137;
hence contradiction by A100,NAT_1:13;
end;
suppose
D2.ID2 in rng D1;
then consider n such that
A101: n in dom D1 and
A102: D1.n=D2.ID2 by PARTFUN1:3;
D2.indx(D2,D1,j1)<D2.ID2 by A94,A95,XXREAL_0:2;
then j1<n by A8,A91,A101,A102,SEQ_4:137;
then
A103: j<n+1 by XREAL_1:19;
n<j by A5,A89,A93,A101,A102,SEQ_4:137;
hence contradiction by A103,NAT_1:13;
end;
end;
D2.ID1 in rng D2 by A92,FUNCT_1:def 3;
then D2.ID1 in {x} by A4,A96,XBOOLE_0:def 3;
then
A104: D2.ID1 = x by TARSKI:def 1;
D2.ID2 in rng D2 by A88,FUNCT_1:def 3;
then D2.ID2 in {x} by A4,A96,XBOOLE_0:def 3;
then D2.ID1=D2.ID2 by A104,TARSKI:def 1;
hence contradiction by A84,A92,A88,SEQ_4:138;
end;
A105: j <= len lower_volume(g,D1) by INTEGRA1:def 7;
A106: 1 <= j by A69,FINSEQ_1:1;
then
A107: mid(lower_volume(g,D1),j,j).1 = lower_volume(g,D1).j by A105,FINSEQ_6:118
;
reconsider lv = lower_volume(g,D1).j as Element of REAL by XREAL_0:def 1;
j-'j+1 = 1 by Lm1;
then len mid(lower_volume(g,D1),j,j)=1 by A106,A105,FINSEQ_6:118;
then
A108: mid(lower_volume(g,D1),j,j)=<*lv*> by A107,FINSEQ_1:40;
A109: 1 <= indx(D2,D1,j1)+1 by A14,NAT_1:13;
indx(D2,D1,j) in dom D2 by A3,A5,INTEGRA1:def 19;
then
A110: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3;
then
A111: 1 <= indx(D2,D1,j) by FINSEQ_1:1;
indx(D2,D1,j) in Seg len lower_volume(g,D2) by A110,INTEGRA1:def 7;
then
A112: indx(D2,D1,j) <= len lower_volume(g,D2) by FINSEQ_1:1;
then
A113: indx(D2,D1,j1)+1 <= len lower_volume(g,D2 ) by A79,XXREAL_0:2;
then indx(D2,D1,j1)+1 in Seg len lower_volume(g,D2) by A109,FINSEQ_1:1;
then
A114: indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 7;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1) =indx(D2,D1,j)-(indx(D2,D1,j1)+1)
by A79,XREAL_1:233;
then indx(D2,D1,j)-'(indx(D2,D1,j1) +1)+1 <= 2 by A82;
then
A115: len mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))<=2 by A79
,A111,A112,A109,A113,FINSEQ_6:118;
len lower_volume(g,D2) = len D2 by INTEGRA1:def 7;
then
A116: indx(D2,D1,j1)+1 in dom D2 by A109,A113,FINSEQ_3:25;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 0+1 by XREAL_1:6;
then
A117: 1 <= len mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) by A79
,A111,A112,A109,A113,FINSEQ_6:118;
now
per cases by A117,A115,Lm2;
suppose
A118: len mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))
=1;
upper_bound divset(D1,j)=D1.j by A5,A7,INTEGRA1:def 4;
then
A119: upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A3,A5,INTEGRA1:def 19;
lower_bound divset(D1,j)=D1.j1 by A5,A7,INTEGRA1:def 4;
then lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A3,A8,
INTEGRA1:def 19;
then
A120: divset(D1,j)=[. D2.indx(D2,D1,j1),D2.indx(D2,D1,j).] by A119,INTEGRA1:4
;
A121: delta(D1) >= 0 by Th9;
A122: upper_bound rng g - lower_bound rng g >= 0 by A72,Lm3,XREAL_1:48;
A123: indx(D2,D1,j) in dom D2 by A3,A5,INTEGRA1:def 19;
len mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 by A79,A111,A112,A109,A113,FINSEQ_6:118;
then
A124: indx(D2,D1,j)-(indx(D2,D1,j1)+1)=0 by A79,A118,XREAL_1:233;
then indx(D2,D1,j)<>1 by A13,FINSEQ_3:25;
then
A125: upper_bound divset(D2,indx(D2,D1,j))=D2.indx(D2,D1,j) by A123,
INTEGRA1:def 4;
lower_bound divset(D2,indx(D2,D1,j))=D2.(indx(D2,D1,j)-1) by A14,A124
,A123,INTEGRA1:def 4;
then
A126: divset(D2,indx(D2,D1,j))=divset(D1,j) by A124,A120,A125,INTEGRA1:4;
reconsider li = lower_volume(g,D2).(indx(D2,D1,j1)+1)
as Element of REAL
by XREAL_0:def 1;
mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).1 =
lower_volume(g,D2).(indx(D2,D1,j1)+1) by A111,A112,A109,A113,FINSEQ_6:118;
then mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =
<*li*> by A118,FINSEQ_1:40;
then Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j )) =
lower_volume(g,D2).(indx(D2,D1,j1)+1) by FINSOP_1:11
.=(lower_bound(rng(g|divset(D2,(indx(D2,D1,j1)+1))))) *vol(divset(
D2,(indx(D2,D1,j1)+1))) by A116,INTEGRA1:def 7
.=lower_volume(g,D1).j by A5,A124,A126,INTEGRA1:def 7
.=Sum mid(lower_volume(g,D1),j,j) by A108,FINSOP_1:11;
hence thesis by A121,A122;
end;
suppose
A127: len mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))
=2;
A128: mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).1 =
lower_volume(g,D2).(indx(D2,D1,j1)+1) by A111,A112,A109,A113,FINSEQ_6:118;
A129: 2+(indx(D2,D1,j1)+1)>=0+1 by XREAL_1:7;
mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).2 =LVg(
D2).(2+(indx(D2,D1,j1)+1)-'1) by A79,A111,A112,A109,A113,A127,FINSEQ_6:118
.=LVg(D2).(2+(indx(D2,D1,j1)+1)-1) by A129,XREAL_1:233
.=LVg(D2).(indx(D2,D1,j1)+(1+1));
then mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =<*
lower_volume(g,D2).(indx(D2,D1,j1)+1), lower_volume(g,D2).(indx(D2,D1,j1)+2)*>
by A127,A128,FINSEQ_1:44;
then
A130: Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j )) =
lower_volume(g,D2).(indx(D2,D1,j1)+1) +lower_volume(g,D2).(indx(D2,D1,j1)+2)
by RVSUM_1:77;
A131: vol(divset(D2,indx(D2,D1,j1)+1))>=0 by INTEGRA1:9;
upper_bound divset(D1,j)=D1.j by A5,A7,INTEGRA1:def 4;
then
A132: upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A3,A5,INTEGRA1:def 19;
A133: vol(divset(D2,indx(D2,D1,j1)+2))>=0 by INTEGRA1:9;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A79,A111,A112,A109,A113,A127,
FINSEQ_6:118;
then
A134: indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A79,XREAL_1:233;
then
A135: indx(D2,D1,j1)+2 in dom D2 by A3,A5,INTEGRA1:def 19;
lower_bound divset(D1,j)=D1.j1 by A5,A7,INTEGRA1:def 4;
then lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A3,A8,
INTEGRA1:def 19;
then
A136: vol(divset(D1,j))=D2.(indx(D2,D1,j1)+2)-D2.(indx(D2,D1,j1)+1) +
D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A132,A134,INTEGRA1:def 5;
indx(D2,D1,j1)+1 in Seg len lower_volume(g,D2) by A109,A113,FINSEQ_1:1;
then indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 7;
then
A137: indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3;
A138: indx(D2,D1,j1)+1 <> 1 by A14,NAT_1:13;
then
A139: upper_bound divset(D2,(indx( D2,D1,j1)+1))=D2.(indx(D2,D1,j1)+1)
by A137,INTEGRA1:def 4;
indx(D2,D1,j1)+1-1=indx(D2,D1,j1)+0;
then
A140: lower_bound divset(D2,(indx(D2,D1,j1)+1))=D2.indx(D2,D1,j1) by A137
,A138,INTEGRA1:def 4;
A141: indx(D2,D1,j1)+1+1 > 1 by A109,NAT_1:13;
indx(D2,D1,j1)+2-1=indx(D2,D1,j1)+1;
then
A142: lower_bound divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+1)
by A135,A141,INTEGRA1:def 4;
upper_bound divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+2)
by A135,A141,INTEGRA1:def 4;
then vol(divset(D1,j))=vol(divset(D2,indx(D2,D1,j1)+2)) +D2.(indx(D2,
D1,j1)+1)-D2.indx(D2,D1,j1) by A142,A136,INTEGRA1:def 5
.=vol(divset(D2,indx(D2,D1,j1)+2)) +(upper_bound divset(D2,indx(D2
,D1,j1)+1)- lower_bound divset(D2,indx(D2,D1,j1)+1)) by A140,A139;
then
A143: vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(
D2,indx(D2,D1,j1)+2)) by INTEGRA1:def 5;
then
A144: lower_volume(g,D1).j=(lower_bound(rng(g|divset(D1,j))))* (vol(
divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2))) by A5,
INTEGRA1:def 7;
A145: Sum mid(LVg(D2),indx(D2,D1,j1)+1,indx(D2,D1,j))-Sum mid(LVg(D1),
j,j) <=(upper_bound rng g - lower_bound rng g)* (vol(divset(D2,indx(D2,D1,j1)+2
)) +vol(divset(D2,indx(D2,D1,j1)+1)))
proof
set ID1=indx(D2,D1,j1)+1, ID2=indx(D2,D1,j1)+2;
set IR = (lower_bound rng g)*vol(divset(D2,ID2));
divset(D1,j)c=A by A5,INTEGRA1:8;
then
A146: lower_bound rng(g|divset(D1,j)) >= lower_bound rng g by A72,Lm4;
Sum mid(LVg(D1),j,j)=(lower_bound rng(g|divset(D1,j)))* vol(
divset(D2,indx(D2, D1,j1)+2)) +(lower_bound rng(g|divset(D1,j)))*vol(divset(D2
,indx(D2,D1,j1)+1)) by A108,A144,FINSOP_1:11;
then Sum mid(LVg(D1),j,j)-(lower_bound rng(g|divset(D1,j)))*vol(
divset(D2,ID1)) >=IR by A133,A146,XREAL_1:64;
then Sum mid(LVg(D1),j,j)>=(lower_bound rng(g|divset(D1,j)))* vol(
divset(D2,ID1)) +IR by XREAL_1:19;
then
A147: Sum mid(LVg(D1),j,j)-(lower_bound rng g)*vol(divset(D2,ID2))
>=(lower_bound rng(g|divset(D1,j)))*vol(divset(D2,ID1)) by XREAL_1:19;
(lower_bound rng(g|divset(D1,j)))*vol(divset(D2,ID1)) >=(
lower_bound rng g)*vol(divset(D2,ID1)) by A131,A146,XREAL_1:64;
then Sum mid(LVg(D1),j,j)-(lower_bound rng g)*vol(divset(D2,ID2))
>=(lower_bound rng g)*vol(divset(D2,ID1)) by A147,XXREAL_0:2;
then
A148: Sum mid(LVg(D1),j,j) >=IR+(lower_bound rng g)* vol(divset(D2,
ID1)) by XREAL_1:19;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A79,A111,A112,A109,A113,A127
,FINSEQ_6:118;
then
A149: indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A79,XREAL_1:233;
ID1 in dom D2 by A114,FINSEQ_1:def 3;
then divset(D2,ID1) c= A by INTEGRA1:8;
then lower_bound rng(g|divset(D2,ID1)) <= upper_bound rng g by A72
,Lm4;
then
A150: (lower_bound rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) <=(
upper_bound rng g)*vol(divset(D2,ID1)) by A131,XREAL_1:64;
A151: indx(D2,D1,j) in dom D2 by A3,A5,INTEGRA1:def 19;
then divset(D2,ID2)c=A by A149,INTEGRA1:8;
then
A152: lower_bound rng(g|divset(D2,ID2)) <= upper_bound rng g by A72,Lm4;
Sum mid(LVg(D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =(
lower_bound rng(g|divset(D2,indx(D2,D1,j1)+2))) *vol(divset(D2,indx(D2,D1,j1)+2
)) +LVg(D2).(indx(D2,D1,j1)+1) by A130,A151,A149,INTEGRA1:def 7
.=(lower_bound rng(g|divset(D2,indx(D2,D1,j1)+2))) *vol(divset(
D2,indx(D2,D1,j1)+2)) +(lower_bound rng(g|divset(D2,indx(D2,D1,j1)+1))) *vol(
divset(D2,indx(D2,D1,j1)+1)) by A116,INTEGRA1:def 7;
then Sum mid(LVg(D2),ID1,indx(D2,D1,j)) -(lower_bound rng(g|divset(
D2,ID1)))*vol(divset(D2,ID1)) <=(upper_bound rng g)*vol(divset(D2,ID2)) by A133
,A152,XREAL_1:64;
then Sum mid(LVg(D2),ID1,indx(D2,D1,j)) <=(upper_bound rng g)*vol(
divset(D2,ID2)) +(lower_bound rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) by
XREAL_1:20;
then Sum mid(LVg(D2),ID1,indx(D2,D1,j))-(upper_bound rng g)*vol(
divset(D2,ID2)) <=(lower_bound rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) by
XREAL_1:20;
then Sum mid(LVg(D2),ID1,indx(D2,D1,j))-(upper_bound rng g)*vol(
divset(D2,ID2)) <=(upper_bound rng g)*vol(divset(D2,ID1)) by A150,XXREAL_0:2;
then Sum mid(LVg(D2),ID1,indx(D2,D1,j))<= (upper_bound rng g)*vol(
divset(D2,ID2)) + (upper_bound rng g)*vol(divset(D2,ID1)) by XREAL_1:20;
then Sum mid(LVg(D2),ID1,indx(D2,D1,j))-Sum mid(LVg(D1),j,j) <=(
upper_bound rng g)*vol(divset(D2,ID2))+(upper_bound rng g)* vol(divset(D2,ID1))
-(IR+(lower_bound rng g)*vol(divset(D2,ID1))) by A148,XREAL_1:13;
hence thesis;
end;
upper_bound rng g - lower_bound rng g >= 0 by A72,Lm3,XREAL_1:48;
then (upper_bound rng g - lower_bound rng g)*(vol(divset(D1,j))) <=(
upper_bound rng g - lower_bound rng g)*delta(D1) by A5,Lm5,XREAL_1:64;
hence thesis by A143,A145,XXREAL_0:2;
end;
end;
hence thesis;
end;
indx(D2,D1,j1) in dom D2 by A3,A12,INTEGRA1:def 19;
then indx(D2,D1,j1) <= len D2 by FINSEQ_3:25;
then
A153: indx(D2,D1,j1) <= len lower_volume(g,D2) by INTEGRA1:def 7;
j1 <= len D1 by A12,FINSEQ_3:25;
then
A154: j1 <= len lower_volume(g,D1) by INTEGRA1:def 7;
A155: D2.indx(D2,D1,j)=D1.(len D1) by A3,A5,INTEGRA1:def 19;
A156: indx(D2,D1,j) >= len lower_volume(g,D2)
proof
assume indx(D2,D1,j) < len lower_volume(g,D2);
then indx(D2,D1,j) < len D2 by INTEGRA1:def 7;
then
A157: D1.(len D1) < D2.(len D2) by A66,A6,A155,SEQM_3:def 1;
A158: not D2.(len D2) in rng D1
proof
assume D2.(len D2) in rng D1;
D2.(len D2) <= upper_bound A by INTEGRA1:def 2;
hence contradiction by A157,INTEGRA1:def 2;
end;
D2.(len D2) in rng D2 by A66,FUNCT_1:def 3;
then D2.(len D2) in rng D1 or D2.(len D2) in {x} by A4,XBOOLE_0:def 3;
then D2.(len D2) = x by A158,TARSKI:def 1;
then D2.(len D2) <= upper_bound divset(D1,len D1) by A1,INTEGRA2:1;
hence contradiction by A5,A7,A157,INTEGRA1:def 4;
end;
indx(D2,D1,j) in Seg len D2 by A6,FINSEQ_1:def 3;
then indx(D2,D1,j) in Seg len lower_volume(g,D2) by INTEGRA1:def 7;
then indx(D2,D1,j) in dom lower_volume(g,D2) by FINSEQ_1:def 3;
then
A159: PLg(D2,indx(D2,D1,j))=Sum(lower_volume(g,D2)|indx(D2,D1,j)) by
INTEGRA1:def 20
.=Sum lower_volume(g,D2) by A156,FINSEQ_1:58;
j in Seg len lower_volume(g,D1) by A69,INTEGRA1:def 7;
then j in dom lower_volume(g,D1) by FINSEQ_1:def 3;
then
A160: PLg(D1,j)=Sum(lower_volume(g,D1)|j) by INTEGRA1:def 20
.=Sum lower_volume(g,D1) by A67,FINSEQ_1:58;
len D1 = len lower_volume(g,D1) by INTEGRA1:def 7;
then
A161: j1 in dom lower_volume(g,D1) by A8,FINSEQ_3:29;
len (D2|indx(D2,D1,j1))=len (D1|j1) by A15,A11,A18,Th6;
then indx(D2,D1,j1) =j1 by A9,A17,FINSEQ_1:59;
then len(lower_volume(g,D1)|j1)=indx(D2,D1,j1) by A154,FINSEQ_1:59;
then len(lower_volume(g,D1)|j1)=len(lower_volume(g,D2)|indx(D2,D1,j1)) by
A153,FINSEQ_1:59;
then
A162: lower_volume(g,D2)|indx(D2,D1,j1)=lower_volume(g,D1)|j1 by A38,
FINSEQ_1:14;
len D2 = len lower_volume(g,D2) by INTEGRA1:def 7;
then indx(D2,D1,j1) in dom lower_volume(g,D2) by A13,FINSEQ_3:29;
then
A163: PLg(D2,indx(D2,D1,j1))=Sum(lower_volume(g,D2)|indx(D2,D1,j1)) by
INTEGRA1:def 20
.=PLg(D1,j1) by A162,A161,INTEGRA1:def 20;
indx(D2,D1,j) <= len D2 by A76,FINSEQ_1:1;
then
A164: indx(D2,D1,j) <= len LVg(D2) by INTEGRA1:def 7;
A165: len D2 = len LVg(D2) by INTEGRA1:def 7;
then
A166: indx(D2,D1,j) in dom LVg(D2) by A75,FINSEQ_3:29;
indx(D2,D1,j1) in dom LVg(D2) by A13,A165,FINSEQ_3:29;
then PLg(D2,indx(D2,D1,j1)) =Sum(LVg(D2)|indx(D2,D1,j1)) by INTEGRA1:def 20;
then PLg(D2,indx(D2,D1,j1)) +Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),
indx(D2,D1,j)) =Sum(LVg(D2)|indx(D2,D1,j1) ^mid(LVg(D2),(indx(D2,D1,j1)+1),indx
(D2,D1,j))) by RVSUM_1:75
.=Sum(mid(LVg(D2),1,indx(D2,D1,j1)) ^mid(LVg(D2),indx(D2,D1,j1)+1,indx(
D2,D1,j))) by A14,FINSEQ_6:116
.=Sum(mid(LVg(D2),1,indx(D2,D1,j))) by A14,A78,A164,INTEGRA2:4
.=Sum(LVg(D2)|indx(D2,D1,j)) by A77,FINSEQ_6:116;
then PLg(D2,indx(D2,D1,j1)) +Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),
indx(D2,D1,j)) =PLg(D2,indx(D2,D1,j)) by A166,INTEGRA1:def 20;
hence thesis by A163,A81,A74,A159,A160;
end;
theorem Th11:
x in divset(D1,len D1) & len D1 >= 2 & D1<=D2 & rng D2 = rng D1
\/ {x} & g|A is bounded implies Sum upper_volume(g,D1) - Sum upper_volume(g,D2)
<= (upper_bound rng g-lower_bound rng g)*delta(D1)
proof
assume that
A1: x in divset(D1,len D1) and
A2: len D1 >= 2;
set j = len D1;
assume that
A3: D1<=D2 and
A4: rng D2 = rng D1 \/ {x};
A5: len D1 in Seg len D1 by FINSEQ_1:3;
then
A6: len D1 in dom D1 by FINSEQ_1:def 3;
then
A7: indx(D2,D1,j) in dom D2 by A3,INTEGRA1:def 19;
deffunc UVg(Division of A) = upper_volume(g,$1);
deffunc PUg(Division of A,Nat) = (PartSums(upper_volume(g,$1))).$2;
A8: j >= len upper_volume(g,D1) by INTEGRA1:def 6;
A9: len D1 <> 1 by A2;
then
A10: len D1-1 in dom D1 by A6,INTEGRA1:7;
reconsider j1=len D1-1 as Element of NAT by A6,A9,INTEGRA1:7;
A11: indx(D2,D1,j1) in dom D2 by A3,A10,INTEGRA1:def 19;
then
A12: 1 <= indx(D2,D1,j1) by FINSEQ_3:25;
then mid(D2,1,indx(D2,D1,j1)) is increasing by A11,INTEGRA1:35;
then
A13: D2|indx(D2,D1,j1) is increasing by A12,FINSEQ_6:116;
len D1 < len D1+1 by NAT_1:13;
then j1 < len D1 by XREAL_1:19;
then
A14: indx(D2,D1,j1) < indx(D2,D1,len D1) by A3,A6,A10,Th8;
then
A15: indx(D2,D1,j1)+1 <= indx(D2,D1,len D1) by NAT_1:13;
len D2 in Seg len D2 by FINSEQ_1:3;
then
A16: len D2 in dom D2 by FINSEQ_1:def 3;
A17: D2.indx(D2,D1,j)=D1.(len D1) by A3,A6,INTEGRA1:def 19;
A18: indx(D2,D1,j) >= len upper_volume(g,D2)
proof
assume indx(D2,D1,j) < len upper_volume(g,D2);
then indx(D2,D1,j) < len D2 by INTEGRA1:def 6;
then
A19: D1.(len D1) < D2.(len D2) by A16,A7,A17,SEQM_3:def 1;
A20: not D2.(len D2) in rng D1
proof
assume D2.(len D2) in rng D1;
D2.(len D2) <= upper_bound A by INTEGRA1:def 2;
hence contradiction by A19,INTEGRA1:def 2;
end;
D2.(len D2) in rng D2 by A16,FUNCT_1:def 3;
then D2.(len D2) in rng D1 or D2.(len D2) in {x} by A4,XBOOLE_0:def 3;
then D2.(len D2) = x by A20,TARSKI:def 1;
then D2.(len D2) <= upper_bound divset(D1,len D1) by A1,INTEGRA2:1;
hence contradiction by A6,A9,A19,INTEGRA1:def 4;
end;
indx(D2,D1,j) in Seg len D2 by A7,FINSEQ_1:def 3;
then indx(D2,D1,j) in Seg len upper_volume(g,D2) by INTEGRA1:def 6;
then indx(D2,D1,j) in dom upper_volume(g,D2) by FINSEQ_1:def 3;
then
A21: PUg(D2,indx(D2,D1,j))=Sum(upper_volume(g,D2)|indx(D2,D1,j)) by
INTEGRA1:def 20
.=Sum upper_volume(g,D2) by A18,FINSEQ_1:58;
indx(D2,D1,j) in dom D2 by A3,A6,INTEGRA1:def 19;
then
A22: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3;
then
A23: 1 <= indx(D2,D1,j) by FINSEQ_1:1;
A24: indx(D2,D1,j1) <= len D2 by A11,FINSEQ_3:25;
then
A25: len (D2|indx(D2,D1,j1))=indx(D2,D1,j1) by FINSEQ_1:59;
A26: j1 <= len D1 by A10,FINSEQ_3:25;
assume
A27: g|A is bounded;
A28: Sum mid(upper_volume(g,D1),len D1,len D1) -Sum mid(upper_volume(g,D2),
(indx(D2,D1,j1)+1),indx(D2,D1,len D1)) <= (upper_bound rng g-lower_bound rng g)
*delta(D1)
proof
A29: indx(D2,D1,j)-indx(D2,D1,j1) <= 2
proof
reconsider ID1=indx(D2,D1,j1)+1 as Element of NAT;
reconsider ID2=ID1+1 as Element of NAT;
assume indx(D2,D1,j)-indx(D2,D1,j1) > 2;
then
A30: indx(D2,D1,j1)+(1+1)<indx(D2,D1,j) by XREAL_1:20;
A31: ID1 < ID2 by NAT_1:13;
then indx(D2,D1,j1) <= ID2 by NAT_1:13;
then
A32: 1 <= ID2 by A12,XXREAL_0:2;
A33: indx(D2,D1,j) in dom D2 by A3,A6,INTEGRA1:def 19;
then
A34: indx(D2,D1,j) <= len D2 by FINSEQ_3:25;
then ID2 <= len D2 by A30,XXREAL_0:2;
then
A35: ID2 in dom D2 by A32,FINSEQ_3:25;
then
A36: D2.ID2<D2.indx(D2, D1,j) by A30,A33,SEQM_3:def 1;
A37: 1 <= ID1 by A12,NAT_1:13;
A38: D1.j1 = D2.indx(D2,D1,j1) by A3,A10,INTEGRA1:def 19;
ID1 <= indx(D2,D1,j) by A30,A31,XXREAL_0:2;
then ID1 <= len D2 by A34,XXREAL_0:2;
then
A39: ID1 in dom D2 by A37,FINSEQ_3:25;
A40: D1.j = D2.indx(D2,D1,j) by A3,A6,INTEGRA1:def 19;
indx(D2,D1,j1) < ID1 by NAT_1:13;
then
A41: D2.indx(D2,D1,j1)<D2.ID1 by A11,A39,SEQM_3:def 1;
A42: D2.ID1<D2.ID2 by A31,A39,A35,SEQM_3:def 1;
A43: not D2.ID1 in rng D1 & not D2.ID2 in rng D1
proof
assume
A44: D2.ID1 in rng D1 or D2.ID2 in rng D1;
now
per cases by A44;
suppose
D2.ID1 in rng D1;
then consider n such that
A45: n in dom D1 and
A46: D1.n=D2.ID1 by PARTFUN1:3;
j1<n by A10,A41,A38,A45,A46,SEQ_4:137;
then
A47: j<n+1 by XREAL_1:19;
D2.ID1<D2.indx(D2,D1,j) by A42,A36,XXREAL_0:2;
then n<j by A6,A40,A45,A46,SEQ_4:137;
hence contradiction by A47,NAT_1:13;
end;
suppose
D2.ID2 in rng D1;
then consider n such that
A48: n in dom D1 and
A49: D1.n=D2.ID2 by PARTFUN1:3;
D2.indx(D2,D1,j1)<D2.ID2 by A41,A42,XXREAL_0:2;
then j1<n by A10,A38,A48,A49,SEQ_4:137;
then
A50: j<n+1 by XREAL_1:19;
n<j by A6,A36,A40,A48,A49,SEQ_4:137;
hence contradiction by A50,NAT_1:13;
end;
end;
hence thesis;
end;
D2.ID1 in rng D2 by A39,FUNCT_1:def 3;
then D2.ID1 in {x} by A4,A43,XBOOLE_0:def 3;
then
A51: D2.ID1 = x by TARSKI:def 1;
D2.ID2 in rng D2 by A35,FUNCT_1:def 3;
then D2.ID2 in {x} by A4,A43,XBOOLE_0:def 3;
then D2.ID1=D2.ID2 by A51,TARSKI:def 1;
hence contradiction by A31,A39,A35,SEQ_4:138;
end;
A52: j <= len upper_volume(g,D1) by INTEGRA1:def 6;
A53: 1 <= j by A5,FINSEQ_1:1;
then
A54: mid(upper_volume(g,D1),j,j).1 = upper_volume(g,D1).j by A52,FINSEQ_6:118;
reconsider uv = upper_volume(g,D1).j as Element of REAL by XREAL_0:def 1;
j-'j+1 = 1 by Lm1;
then len mid(upper_volume(g,D1),j,j)=1 by A53,A52,FINSEQ_6:118;
then mid(upper_volume(g,D1),j,j)=<*uv*> by A54,FINSEQ_1:40;
then
A55: Sum mid(upper_volume(g,D1),j,j)=upper_volume(g,D1).j by FINSOP_1:11;
A56: 1 <= indx(D2,D1,j1)+1 by A12,NAT_1:13;
indx(D2,D1,j) in dom D2 by A3,A6,INTEGRA1:def 19;
then
A57: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3;
then
A58: 1 <= indx(D2,D1,j) by FINSEQ_1:1;
indx(D2,D1,j) in Seg len upper_volume(g,D2) by A57,INTEGRA1:def 6;
then
A59: indx(D2,D1,j) <= len upper_volume(g,D2) by FINSEQ_1:1;
then
A60: indx(D2,D1,j1)+1 <= len upper_volume(g,D2 ) by A15,XXREAL_0:2;
then indx(D2,D1,j1)+1 in Seg len upper_volume(g,D2) by A56,FINSEQ_1:1;
then
A61: indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 6;
then
A62: indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1) =indx(D2,D1,j)-(indx(D2,D1,j1)+1)
by A15,XREAL_1:233;
then indx(D2,D1,j)-'(indx(D2,D1,j1) +1)+1 <= 2 by A29;
then
A63: len mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))<=2 by A15,A58
,A59,A56,A60,FINSEQ_6:118;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 0+1 by XREAL_1:6;
then
A64: 1 <= len mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) by A15
,A58,A59,A56,A60,FINSEQ_6:118;
now
per cases by A64,A63,Lm2;
suppose
A65: len mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))
=1;
upper_bound divset(D1,j)=D1.j by A6,A9,INTEGRA1:def 4;
then
A66: upper_bound divset(D1,j)=D2.indx(D2,D1,j) by A3,A6,INTEGRA1:def 19;
lower_bound divset(D1,j)=D1.j1 by A6,A9,INTEGRA1:def 4;
then lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A3,A10,
INTEGRA1:def 19;
then
A67: divset(D1,j)=[. D2.indx(D2,D1,j1),D2.indx(D2,D1,j).] by A66,INTEGRA1:4;
A68: delta(D1) >= 0 by Th9;
A69: upper_bound rng g - lower_bound rng g >= 0 by A27,Lm3,XREAL_1:48;
A70: indx(D2,D1,j) in dom D2 by A3,A6,INTEGRA1:def 19;
len mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 by A15,A58,A59,A56,A60,FINSEQ_6:118
;
then
A71: indx(D2,D1,j)-(indx(D2,D1,j1)+1)=0 by A15,A65,XREAL_1:233;
then indx(D2,D1,j)<>1 by A11,FINSEQ_3:25;
then
A72: upper_bound divset(D2,indx(D2,D1,j))=D2.indx(D2,D1,j) by A70,
INTEGRA1:def 4;
lower_bound divset(D2,indx(D2,D1,j))=D2.(indx(D2,D1,j)-1) by A12,A71
,A70,INTEGRA1:def 4;
then
A73: divset(D2,indx(D2,D1,j))=divset(D1,j) by A71,A67,A72,INTEGRA1:4;
reconsider uv =upper_volume(g,D2).(indx(D2,D1,j1)+1)
as Element of REAL
by XREAL_0:def 1;
mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).1 =
upper_volume(g,D2).(indx(D2,D1,j1)+1) by A58,A59,A56,A60,FINSEQ_6:118;
then mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))
= <*uv*> by A65,FINSEQ_1:40;
then Sum mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j )) =
upper_volume(g,D2).(indx(D2,D1,j1)+1) by FINSOP_1:11
.=(upper_bound(rng(g|divset(D2,(indx(D2,D1,j1)+1))))) *vol(divset(
D2,(indx(D2,D1,j1)+1))) by A62,INTEGRA1:def 6
.=Sum mid(upper_volume(g,D1),j,j) by A6,A55,A71,A73,INTEGRA1:def 6;
hence thesis by A68,A69;
end;
suppose
A74: len mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))
=2;
A75: mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).1 =
upper_volume(g,D2).(indx(D2,D1,j1)+1) by A58,A59,A56,A60,FINSEQ_6:118;
A76: 2+(indx(D2,D1,j1)+1)>=0+1 by XREAL_1:7;
mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).2 =UVg(
D2).(2+(indx(D2,D1,j1)+1)-'1) by A15,A58,A59,A56,A60,A74,FINSEQ_6:118
.=UVg(D2).(2+(indx(D2,D1,j1)+1)-1) by A76,XREAL_1:233
.=UVg(D2).(indx(D2,D1,j1)+(1+1));
then mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =<*
upper_volume(g,D2).(indx(D2,D1,j1)+1), upper_volume(g,D2).(indx(D2,D1,j1)+2)*>
by A74,A75,FINSEQ_1:44;
then
A77: Sum mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j )) =
upper_volume(g,D2).(indx(D2,D1,j1)+1) +upper_volume(g,D2).(indx(D2,D1,j1)+2)
by RVSUM_1:77;
A78: vol(divset(D2,indx(D2,D1,j1)+1))>=0 by INTEGRA1:9;
upper_bound divset(D1,j)=D1.j by A6,A9,INTEGRA1:def 4;
then
A79: upper_bound divset(D1,j) =D2. indx(D2,D1,j) by A3,A6,INTEGRA1:def 19;
A80: vol(divset(D2,indx(D2,D1,j1)+2))>=0 by INTEGRA1:9;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A15,A58,A59,A56,A60,A74,
FINSEQ_6:118;
then
A81: indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A15,XREAL_1:233;
then
A82: indx(D2,D1,j1)+2 in dom D2 by A3,A6,INTEGRA1:def 19;
lower_bound divset(D1,j)=D1.j1 by A6,A9,INTEGRA1:def 4;
then lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A3,A10,
INTEGRA1:def 19;
then
A83: vol(divset(D1,j)) =D2.(indx(D2,D1,j1)+2)-D2.(indx(D2,D1,j1)+1) +
D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A79,A81,INTEGRA1:def 5;
indx(D2,D1,j1)+1 in Seg len upper_volume(g,D2) by A56,A60,FINSEQ_1:1;
then indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 6;
then
A84: indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3;
A85: indx(D2,D1,j1)+1 <> 1 by A12,NAT_1:13;
then
A86: upper_bound divset(D2,(indx( D2,D1,j1)+1))=D2.(indx(D2,D1,j1)+1)
by A84,INTEGRA1:def 4;
indx(D2,D1,j1)+1-1=indx(D2,D1,j1)+0;
then
A87: lower_bound divset(D2,(indx(D2,D1,j1)+1))=D2.indx(D2,D1,j1) by A84,A85,
INTEGRA1:def 4;
A88: indx(D2,D1,j1)+1+1 > 1 by A56,NAT_1:13;
indx(D2,D1,j1)+2-1=indx(D2,D1,j1)+1;
then
A89: lower_bound divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+1)
by A82,A88,INTEGRA1:def 4;
upper_bound divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+2)
by A82,A88,INTEGRA1:def 4;
then vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+2)) +D2.(indx(D2
,D1,j1)+1)-D2.indx(D2,D1,j1) by A89,A83,INTEGRA1:def 5
.=vol(divset(D2,indx(D2,D1,j1)+2)) +(upper_bound divset(D2,indx(D2
,D1,j1)+1)-lower_bound divset(D2,indx(D2,D1,j1)+1)) by A87,A86;
then
A90: vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(
D2,indx(D2,D1,j1)+2)) by INTEGRA1:def 5;
then
A91: upper_volume(g,D1).j=(upper_bound(rng(g|divset(D1,j))))* (vol(
divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2))) by A6,
INTEGRA1:def 6;
A92: Sum mid(UVg(D1),j,j)-Sum mid(UVg(D2),indx(D2,D1,j1)+1,indx(D2,D1
,j)) <=(upper_bound rng g - lower_bound rng g)*(vol(divset(D2, indx(D2,D1,j1)+2
)) +vol(divset(D2,indx(D2,D1,j1)+1)))
proof
set ID1=indx(D2,D1,j1)+1, ID2=indx(D2,D1,j1)+2;
A93: Sum mid(UVg(D1),j,j)-(upper_bound rng(g|divset(D1,j)))* vol(
divset(D2,ID1)) = (upper_bound rng(g|divset(D1,j)))*vol(divset(D2,ID2)) by A55
,A91;
divset(D1,j)c=A by A6,INTEGRA1:8;
then
A94: upper_bound rng(g|divset(D1,j)) <= upper_bound rng g by A27,Lm4;
then (upper_bound rng(g|divset(D1,j)))*vol(divset(D2,ID2)) <=(
upper_bound rng g)*vol(divset(D2,ID2)) by A80,XREAL_1:64;
then Sum mid(UVg(D1),j,j)<=(upper_bound rng(g|divset(D1,j)))* vol(
divset(D2,ID1)) +(upper_bound rng g)*vol(divset(D2,ID2)) by A93,XREAL_1:20;
then
A95: Sum mid(UVg(D1),j,j)-(upper_bound rng g)*vol(divset(D2,ID2))
<=(upper_bound rng(g|divset(D1,j)))*vol(divset(D2,ID1)) by XREAL_1:20;
(upper_bound rng(g|divset(D1,j)))*vol(divset(D2,ID1)) <=(
upper_bound rng g)*vol(divset(D2,ID1)) by A78,A94,XREAL_1:64;
then Sum mid(UVg(D1),j,j)-(upper_bound rng g)*vol(divset(D2,ID2))
<=(upper_bound rng g)*vol(divset(D2,ID1)) by A95,XXREAL_0:2;
then
A96: Sum mid(UVg(D1),j,j) <=(upper_bound rng g)*vol(divset(D2,ID2))
+(upper_bound rng g)* vol(divset(D2,ID1)) by XREAL_1:20;
ID1 in dom D2 by A61,FINSEQ_1:def 3;
then divset(D2,ID1)c=A by INTEGRA1:8;
then upper_bound rng(g|divset(D2,ID1)) >= lower_bound rng g by A27
,Lm4;
then
A97: (upper_bound rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) >=(
lower_bound rng g)*vol(divset(D2,ID1)) by A78,XREAL_1:64;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A15,A58,A59,A56,A60,A74,
FINSEQ_6:118;
then
A98: indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A15,XREAL_1:233;
A99: indx(D2,D1,j) in dom D2 by A3,A6,INTEGRA1:def 19;
then divset(D2,ID2)c=A by A98,INTEGRA1:8;
then
A100: upper_bound rng(g|divset(D2,ID2)) >= lower_bound rng g by A27,Lm4;
Sum mid(UVg(D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =(
upper_bound rng(g|divset(D2,indx(D2,D1,j1)+2))) *vol(divset(D2,indx(D2,D1,j1)+2
)) +UVg(D2).(indx(D2,D1,j1)+1) by A77,A99,A98,INTEGRA1:def 6
.=(upper_bound rng(g|divset(D2,indx(D2,D1,j1)+2))) *vol(divset(
D2,indx(D2,D1,j1)+2)) +(upper_bound rng(g|divset(D2,indx(D2,D1,j1)+1))) *vol(
divset(D2,indx(D2,D1,j1)+1)) by A62,INTEGRA1:def 6;
then Sum mid(UVg(D2),ID1,indx(D2,D1,j)) -(upper_bound rng(g|divset(
D2,ID1)))*vol(divset(D2,ID1)) >=(lower_bound rng g)*vol(divset(D2,ID2)) by A80
,A100,XREAL_1:64;
then Sum mid(UVg(D2),ID1,indx(D2,D1,j)) >=(lower_bound rng g)*vol(
divset(D2,ID2)) +(upper_bound rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) by
XREAL_1:19;
then Sum mid(UVg(D2),ID1,indx(D2,D1,j))-(lower_bound rng g)*vol(
divset(D2,ID2)) >=(upper_bound rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) by
XREAL_1:19;
then Sum mid(UVg(D2),ID1,indx(D2,D1,j))-(lower_bound rng g)*vol(
divset(D2,ID2)) >=(lower_bound rng g)*vol(divset(D2,ID1)) by A97,XXREAL_0:2;
then Sum mid(UVg(D2),ID1,indx(D2,D1,j)) >= (lower_bound rng g)*vol(
divset(D2,ID2) ) + (lower_bound rng g)*vol(divset(D2,ID1)) by XREAL_1:19;
then Sum mid(UVg(D1),j,j)-Sum mid(UVg(D2),ID1,indx(D2,D1,j)) <=(
upper_bound rng g)*vol(divset(D2,ID2))+(upper_bound rng g)* vol(divset(D2,ID1))
-((lower_bound rng g)*vol(divset(D2,ID2))+(lower_bound rng g)* vol(divset(D2,
ID1))) by A96,XREAL_1:13;
hence thesis;
end;
upper_bound rng g - lower_bound rng g >= 0 by A27,Lm3,XREAL_1:48;
then (upper_bound rng g - lower_bound rng g)*(vol(divset(D1,j))) <=(
upper_bound rng g - lower_bound rng g)*delta(D1) by A6,Lm5,XREAL_1:64;
hence thesis by A90,A92,XXREAL_0:2;
end;
end;
hence thesis;
end;
j in Seg len upper_volume(g,D1) by A5,INTEGRA1:def 6;
then j in dom upper_volume(g,D1) by FINSEQ_1:def 3;
then
A101: PUg(D1,j)=Sum(upper_volume(g,D1)|j) by INTEGRA1:def 20
.=Sum upper_volume(g,D1) by A8,FINSEQ_1:58;
A102: j <= len UVg(D1) by INTEGRA1:def 6;
A103: 1 <= j1 by A10,FINSEQ_3:25;
then mid(D1,1,j1) is increasing by A6,A9,INTEGRA1:7,35;
then
A104: D1|j1 is increasing by A103,FINSEQ_6:116;
A105: rng (D2|indx(D2,D1,j1)) = rng (D1|j1) by A1,A2,A3,A4,Lm6;
then
A106: D2|indx(D2,D1,j1)=D1|j1 by A13,A104,Th6;
A107: for k st 1 <= k & k <= j1 holds k=indx(D2,D1,k)
proof
let k;
assume that
A108: 1 <= k and
A109: k <= j1;
assume
A110: k<>indx(D2,D1,k);
now
per cases by A110,XXREAL_0:1;
suppose
A111: k > indx(D2,D1,k);
k <= len D1 by A26,A109,XXREAL_0:2;
then
A112: k in dom D1 by A108,FINSEQ_3:25;
then indx(D2,D1,k) in dom D2 by A3,INTEGRA1:def 19;
then indx(D2,D1,k) in Seg len D2 by FINSEQ_1:def 3;
then
A113: 1<=indx(D2,D1,k) by FINSEQ_1:1;
A114: indx(D2,D1,k) < j1 by A109,A111,XXREAL_0:2;
then
A115: indx(D2,D1,k) in Seg j1 by A113,FINSEQ_1:1;
indx(D2,D1,k)<=indx(D2,D1,j1) by A3,A10,A109,A112,Th7;
then indx(D2,D1,k) in Seg indx(D2,D1,j1) by A113,FINSEQ_1:1;
then
A116: (D2|indx(D2,D1,j1)).indx(D2,D1,k)=D2.indx(D2,D1,k) by A11,RFINSEQ:6;
indx(D2,D1,k) <= len D1 by A26,A114,XXREAL_0:2;
then indx(D2,D1,k) in Seg len D1 by A113,FINSEQ_1:1;
then indx(D2,D1,k) in dom D1 by FINSEQ_1:def 3;
then
A117: D1.k > D1.indx(D2,D1,k) by A111,A112,SEQM_3:def 1;
D1.k=D2.indx(D2,D1,k) by A3,A112,INTEGRA1:def 19;
hence contradiction by A10,A106,A116,A117,A115,RFINSEQ:6;
end;
suppose
A118: k < indx(D2,D1,k);
k <= len D1 by A26,A109,XXREAL_0:2;
then
A119: k in dom D1 by A108,FINSEQ_3:25;
then indx(D2,D1,k) <= indx(D2,D1,j1) by A3,A10,A109,Th7;
then
A120: k <= indx(D2,D1,j1) by A118,XXREAL_0:2;
then k <= len D2 by A24,XXREAL_0:2;
then
A121: k in dom D2 by A108,FINSEQ_3:25;
k in Seg j1 by A108,A109,FINSEQ_1:1;
then
A122: D1.k = (D1|j1).k by A10,RFINSEQ:6;
indx(D2,D1,k) in dom D2 by A3,A119,INTEGRA1:def 19;
then
A123: D2.k < D2.indx(D2,D1,k) by A118,A121,SEQM_3:def 1;
A124: k in Seg indx(D2,D1,j1) by A108,A120,FINSEQ_1:1;
D1.k=D2.indx(D2,D1,k) by A3,A119,INTEGRA1:def 19;
hence contradiction by A11,A106,A122,A123,A124,RFINSEQ:6;
end;
end;
hence contradiction;
end;
A125: for k be Nat st 1 <= k & k <= len(upper_volume(g,D1)|j1) holds (
upper_volume(g,D1)|j1).k = (upper_volume(g,D2)|indx(D2,D1,j1)).k
proof
indx(D2,D1,j1) in Seg len D2 by A11,FINSEQ_1:def 3;
then indx(D2,D1,j1) in Seg len upper_volume(g,D2) by INTEGRA1:def 6;
then
A126: indx(D2,D1,j1) in dom upper_volume(g,D2) by FINSEQ_1:def 3;
let k be Nat;
assume that
A127: 1 <= k and
A128: k <= len(upper_volume(g,D1)|j1);
reconsider k as Element of NAT by ORDINAL1:def 12;
A129: len(upper_volume(g,D1)) = len D1 by INTEGRA1:def 6;
then
A130: k <= j1 by A26,A128,FINSEQ_1:59;
then
A131: k <= len D1 by A26,XXREAL_0:2;
then k in Seg len D1 by A127,FINSEQ_1:1;
then
A132: k in dom D1 by FINSEQ_1:def 3;
then
A133: indx(D2,D1,k) in dom D2 by A3,INTEGRA1:def 19;
A134: k in Seg j1 by A127,A130,FINSEQ_1:1;
then indx(D2,D1,k) in Seg j1 by A107,A127,A130;
then
A135: indx(D2,D1,k) in Seg indx(D2,D1,j1) by A103,A107;
then indx(D2,D1,k)<=indx(D2,D1,j1) by FINSEQ_1:1;
then
A136: indx(D2,D1,k)<=len D2 by A24,XXREAL_0:2;
A137: D1.k=D2.indx(D2,D1,k) by A3,A132,INTEGRA1:def 19;
A138: lower_bound divset(D1,k)=lower_bound divset(D2,indx(D2,D1,k)) &
upper_bound divset(D1,k)=upper_bound divset(D2,indx(D2,D1,k))
proof
per cases;
suppose
A139: k=1;
then
A140: upper_bound divset(D1,k)=D1.k by A132,INTEGRA1:def 4;
A141: lower_bound divset(D1,k)=lower_bound A by A132,A139,INTEGRA1:def 4;
indx(D2,D1,k)=1 by A103,A107,A139;
hence thesis by A133,A137,A141,A140,INTEGRA1:def 4;
end;
suppose
A142: k<>1;
then reconsider k1=k-1 as Element of NAT by A132,INTEGRA1:7;
k <= k+1 by NAT_1:11;
then k1 <= k by XREAL_1:20;
then
A143: k1 <= j1 by A130,XXREAL_0:2;
A144: k-1 in dom D1 by A132,A142,INTEGRA1:7;
then 1 <= k1 by FINSEQ_3:25;
then k1=indx(D2,D1,k1) by A107,A143;
then
A145: D2.(indx(D2,D1,k)-1)= D2.indx(D2,D1,k1) by A107,A127,A130;
A146: indx(D2,D1,k)<>1 by A107,A127,A130,A142;
then
A147: lower_bound divset(D2,indx(D2,D1,k))=D2.(indx(D2,D1,k)-1) by A133,
INTEGRA1:def 4;
A148: upper_bound divset(D2,indx(D2,D1,k))=D2.indx(D2,D1,k) by A133,A146,
INTEGRA1:def 4;
A149: upper_bound divset(D1,k)=D1.k by A132,A142,INTEGRA1:def 4;
lower_bound divset(D1,k)=D1.(k-1) by A132,A142,INTEGRA1:def 4;
hence thesis by A3,A132,A149,A144,A147,A148,A145,INTEGRA1:def 19;
end;
end;
divset(D1,k)=[. lower_bound divset(D1,k), upper_bound divset(D1,k) .]
by INTEGRA1:4;
then
A150: divset(D1,k)=divset(D2,indx(D2,D1,k)) by A138,INTEGRA1:4;
A151: k in dom D1 by A127,A131,FINSEQ_3:25;
j1 in Seg len(upper_volume(g,D1)) by A10,A129,FINSEQ_1:def 3;
then j1 in dom(upper_volume(g,D1)) by FINSEQ_1:def 3;
then
A152: (upper_volume(g,D1)|j1).k = upper_volume(g,D1).k by A134,RFINSEQ:6
.=(upper_bound(rng(g|divset(D2,indx(D2,D1,k)))))*vol(divset(D2, indx(
D2,D1,k))) by A151,A150,INTEGRA1:def 6;
1<=indx(D2,D1,k) by A107,A127,A130;
then
A153: indx(D2,D1,k) in dom D2 by A136,FINSEQ_3:25;
(upper_volume(g,D2)|indx(D2,D1,j1)).k =(upper_volume(g,D2)|indx(D2,
D1,j1)).indx(D2,D1,k) by A107,A127,A130
.=upper_volume(g,D2).indx(D2,D1,k) by A135,A126,RFINSEQ:6
.=(upper_bound(rng(g|divset(D2,indx(D2,D1,k)))))*vol(divset(D2, indx(
D2,D1,k))) by A153,INTEGRA1:def 6;
hence thesis by A152;
end;
indx(D2,D1,j1) in dom D2 by A3,A10,INTEGRA1:def 19;
then indx(D2,D1,j1) <= len D2 by FINSEQ_3:25;
then
A154: indx(D2,D1,j1) <= len upper_volume(g,D2) by INTEGRA1:def 6;
j1 in Seg len D1 by A10,FINSEQ_1:def 3;
then j1 <= len D1 by FINSEQ_1:1;
then
A155: j1 <= len upper_volume(g,D1) by INTEGRA1:def 6;
len (D2|indx(D2,D1,j1))=len (D1|j1) by A13,A104,A105,Th6;
then indx(D2,D1,j1) =j1 by A26,A25,FINSEQ_1:59;
then len(upper_volume(g,D1)|j1)=indx(D2,D1,j1) by A155,FINSEQ_1:59;
then len(upper_volume(g,D1)|j1)=len(upper_volume(g,D2)|indx(D2,D1,j1)) by
A154,FINSEQ_1:59;
then
A156: upper_volume(g,D2)|indx(D2,D1,j1)=upper_volume(g,D1)|j1 by A125,
FINSEQ_1:14;
j1 in Seg len D1 by A10,FINSEQ_1:def 3;
then j1 in Seg len upper_volume(g,D1) by INTEGRA1:def 6;
then
A157: j1 in dom upper_volume(g,D1) by FINSEQ_1:def 3;
j < j+1 by NAT_1:13;
then
A158: j1 < j by XREAL_1:19;
indx(D2,D1,j) <= len D2 by A22,FINSEQ_1:1;
then
A159: indx(D2,D1,j) <= len UVg(D2) by INTEGRA1:def 6;
then
A160: indx(D2,D1,j) in dom UVg(D2) by A23,FINSEQ_3:25;
indx(D2,D1,j1) in Seg len D2 by A11,FINSEQ_1:def 3;
then indx(D2,D1,j1) in Seg len UVg(D2) by INTEGRA1:def 6;
then indx(D2,D1,j1) in dom UVg(D2) by FINSEQ_1:def 3;
then PUg(D2,indx(D2,D1,j1)) =Sum(UVg(D2)|indx(D2,D1,j1)) by INTEGRA1:def 20;
then PUg(D2,indx(D2,D1,j1)) +Sum mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),
indx(D2,D1,j)) =Sum(UVg(D2)|indx(D2,D1,j1) ^mid(UVg(D2),(indx(D2,D1,j1)+1),indx
(D2,D1,j))) by RVSUM_1:75
.=Sum(mid(UVg(D2),1,indx(D2,D1,j1)) ^mid(UVg(D2),indx(D2,D1,j1)+1,indx(
D2,D1,j))) by A12,FINSEQ_6:116
.=Sum(mid(UVg(D2),1,indx(D2,D1,j))) by A12,A14,A159,INTEGRA2:4
.=Sum(UVg(D2)|indx(D2,D1,j)) by A23,FINSEQ_6:116;
then
A161: PUg(D2,indx(D2,D1,j1)) +Sum mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),
indx(D2,D1,j)) =PUg(D2,indx(D2,D1,j)) by A160,INTEGRA1:def 20;
A162: 1 <= j by A5,FINSEQ_1:1;
then
A163: j in dom UVg(D1) by A102,FINSEQ_3:25;
j1 in Seg len D1 by A10,FINSEQ_1:def 3;
then j1 in Seg len UVg(D1) by INTEGRA1:def 6;
then j1 in dom UVg(D1) by FINSEQ_1:def 3;
then PUg(D1,j1)=Sum(UVg(D1)|j1) by INTEGRA1:def 20;
then PUg(D1,j1)+Sum mid(UVg(D1),j,j) =Sum((UVg(D1)|j1)^mid(UVg(D1),j,j)) by
RVSUM_1:75
.=Sum(mid(UVg(D1),1,j1)^mid(UVg(D1),j1+1,j)) by A103,FINSEQ_6:116
.=Sum(mid(UVg(D1),1,j)) by A103,A102,A158,INTEGRA2:4
.=Sum(UVg(D1)|j) by A162,FINSEQ_6:116;
then
A164: PUg(D1,j1)+Sum mid(upper_volume(g,D1),j,j)=PUg(D1,j) by A163,
INTEGRA1:def 20;
indx(D2,D1,j1) in Seg len D2 by A11,FINSEQ_1:def 3;
then indx(D2,D1,j1) in Seg len upper_volume(g,D2) by INTEGRA1:def 6;
then indx(D2,D1,j1) in dom upper_volume(g,D2) by FINSEQ_1:def 3;
then PUg(D2,indx(D2,D1,j1))=Sum(upper_volume(g,D2)|indx(D2,D1,j1)) by
INTEGRA1:def 20
.=PUg(D1,j1) by A156,A157,INTEGRA1:def 20;
hence thesis by A28,A161,A164,A21,A101;
end;
Lm7: vol(A)<>0 & y in rng lower_sum_set(f) implies ex D being Division of A st
D in dom lower_sum_set(f) & y=(lower_sum_set(f)).D & D.1 > lower_bound A
proof
assume
A1: vol(A)<>0;
assume y in rng lower_sum_set(f);
then consider D3 being Element of divs A such that
A2: D3 in dom lower_sum_set(f) and
A3: y=(lower_sum_set(f)).D3 by PARTFUN1:3;
reconsider D3 as Division of A by INTEGRA1:def 3;
rng D3 <> {};
then
A4: 1 in dom D3 by FINSEQ_3:32;
A5: len D3 in Seg len D3 by FINSEQ_1:3;
now
per cases;
suppose
A6: D3.1 <> lower_bound A;
D3.1 in A by A4,INTEGRA1:6;
then lower_bound A <= D3.1 by INTEGRA2:1;
then D3.1 > lower_bound A by A6,XXREAL_0:1;
hence thesis by A2,A3;
end;
suppose
A7: D3.1 = lower_bound A;
ex D being Division of A st D in dom lower_sum_set(f) & y=(
lower_sum_set(f)).D & D.1 > lower_bound A
proof
A8: lower_volume(f,D3).1=(lower_bound rng(f|divset(D3,1)))* vol(
divset(D3,1)) by A4,INTEGRA1:def 7;
vol(A) >= 0 by INTEGRA1:9;
then
A9: upper_bound A - lower_bound A > 0 by A1,INTEGRA1:def 5;
A10: y=lower_sum(f,D3) by A3,INTEGRA1:def 11
.=Sum(lower_volume(f,D3)) by INTEGRA1:def 9
.=Sum((lower_volume(f,D3)|1)^(lower_volume(f,D3)/^1)) by RFINSEQ:8;
A11: D3.(len D3) = upper_bound A by INTEGRA1:def 2;
len D3 in dom D3 by A5,FINSEQ_1:def 3;
then
A12: len D3 > 1 by A4,A7,A11,A9,SEQ_4:137,XREAL_1:47;
then reconsider D=D3/^1 as increasing FinSequence of REAL by
INTEGRA1:34;
A13: len D = len D3 - 1 by A12,RFINSEQ:def 1;
upper_bound A > lower_bound A by A9,XREAL_1:47;
then len D <> 0 by A7,A13,INTEGRA1:def 2;
then reconsider D as non empty increasing FinSequence of REAL;
A14: len D in dom D by FINSEQ_5:6;
len D+1=len D3 by A13;
then
A15: D.(len D)=upper_bound A by A11,A12,A14,RFINSEQ:def 1;
A16: len D3 >= 1+1 by A12,NAT_1:13;
then
A17: 2 <= len lower_volume(f,D3) by INTEGRA1:def 7;
1+1 <= len D3 by A12,NAT_1:13;
then 2 in dom D3 by FINSEQ_3:25;
then
A18: D3.1 < D3.2 by A4,SEQM_3:def 1;
A19: rng D3 c= A by INTEGRA1:def 2;
rng D c= rng D3 by FINSEQ_5:33;
then rng D c= A by A19;
then reconsider D as Division of A by A15,INTEGRA1:def 2;
A20: 1 in Seg 1 by FINSEQ_1:1;
A21: 1 <= len lower_volume(f,D3) by A12,INTEGRA1:def 7;
A22: len (lower_volume(f,D3)|1)=1;
1 <= len lower_volume(f,D3) by A12,INTEGRA1:def 7;
then
A23: len(mid(lower_volume(f,D3),2,len lower_volume(f,D3))) =len
lower_volume(f,D3)-'2+1 by A17,FINSEQ_6:118
.=len D3-'2+1 by INTEGRA1:def 7
.=len D3-2+1 by A16,XREAL_1:233
.=len D3-1;
A24: for i be Nat st 1<=i&i<=len mid(lower_volume(f,D3),2,len
lower_volume(f,D3)) holds mid(lower_volume(f,D3),2,len lower_volume(f,D3)).i =
lower_volume(f,D).i
proof
let i be Nat;
assume that
A25: 1<=i and
A26: i<=len mid(lower_volume(f,D3),2,len lower_volume(f,D3));
A27: 1 <= i+1 by NAT_1:12;
i+1 <= len D3 by A23,A26,XREAL_1:19;
then
A28: i+1 in Seg len D3 by A27,FINSEQ_1:1;
then
A29: i+1 in dom D3 by FINSEQ_1:def 3;
A30: divset(D3,i+1)=divset(D,i)
proof
A31: i+1 in dom D3 by A28,FINSEQ_1:def 3;
A32: 1<>i+1 by A25,NAT_1:13;
then
A33: upper_bound divset(D3,i+1)=D3.(i+1) by A31,INTEGRA1:def 4;
A34: i in dom D by A13,A23,A25,A26,FINSEQ_3:25;
then
A35: D.i=D3.(i+1) by A12,RFINSEQ:def 1;
A36: lower_bound divset( D3,i+1)=D3.(i+1- 1) by A32,A31,INTEGRA1:def 4;
per cases;
suppose
A37: i=1;
then
A38: upper_bound divset(D,i )=D.i by A34,INTEGRA1:def 4;
A39: lower_bound divset(D,i)=lower_bound A by A34,A37,INTEGRA1:def 4;
divset(D3,i+1)=[.lower_bound A, D.i.] by A7,A33,A36,A35,A37,
INTEGRA1:4;
hence thesis by A39,A38,INTEGRA1:4;
end;
suppose
A40: i<>1;
then i-1 in dom D by A34,INTEGRA1:7;
then
A41: D.(i-1)=D3.(i-1+1) by A12,RFINSEQ:def 1
.=D3.i;
A42: upper_bound divset(D,i)=D.i by A34,A40,INTEGRA1:def 4;
lower_bound divset(D,i)=D.(i-1) by A34,A40,INTEGRA1:def 4;
then divset(D3,i+1)=[.lower_bound divset(D,i), upper_bound
divset(D,i).] by A33,A36,A35,A42,A41,INTEGRA1:4;
hence thesis by INTEGRA1:4;
end;
end;
i<=len lower_volume(f,D3) - 1 by A23,A26,INTEGRA1:def 7;
then
A43: i <= len lower_volume(f,D3)-2+1;
mid(lower_volume(f,D3),2,len lower_volume(f,D3 )).i =
lower_volume(f,D3).(i+2-1) by A17,A25,A43,FINSEQ_6:122
.=lower_volume(f,D3).(i+1);
then
A44: mid(lower_volume(f,D3),2,len lower_volume(f,D3 )).i =(
lower_bound rng(f|divset(D3,i+1)))*vol(divset(D3,i+1)) by A29,INTEGRA1:def 7;
i in Seg len D by A13,A23,A25,A26,FINSEQ_1:1;
then i in dom D by FINSEQ_1:def 3;
hence thesis by A44,A30,INTEGRA1:def 7;
end;
1 in dom lower_volume(f,D3) by A21,FINSEQ_3:25;
then (lower_volume(f,D3)|1).1 = lower_volume(f,D3).1 by A20,RFINSEQ:6;
then
A45: lower_volume(f,D3)|1 = <*lower_volume(f,D3).1*> by A22,FINSEQ_1:40;
A46: 2-'1=2-1 by XREAL_1:233
.= 1;
rng D <> {};
then 1 in dom D by FINSEQ_3:32;
then
A47: D.1=D3.(1+1) by A12,RFINSEQ:def 1
.=D3.2;
D in divs A by INTEGRA1:def 3;
then
A48: D in dom lower_sum_set(f) by FUNCT_2:def 1;
len lower_volume(f,D3) >= 2 by A16,INTEGRA1:def 7;
then
A49: mid(lower_volume(f,D3),2,len lower_volume(f,D3)) =lower_volume(f,
D3)/^1 by A46,FINSEQ_6:117;
len(mid(lower_volume(f,D3),2,len lower_volume(f,D3))) =len
lower_volume(f, D) by A13,A23,INTEGRA1:def 7;
then
A50: lower_volume(f,D3)/^1 = lower_volume(f,D) by A49,A24,FINSEQ_1:14;
vol(divset(D3,1))=upper_bound divset(D3,1)-lower_bound divset(D3,
1) by INTEGRA1:def 5
.=upper_bound divset(D3,1) - lower_bound A by A4,INTEGRA1:def 4
.=D3.1 - lower_bound A by A4,INTEGRA1:def 4
.=0 by A7;
then y=0+Sum(lower_volume(f,D)) by A10,A45,A8,A50,RVSUM_1:76
.=lower_sum(f,D) by INTEGRA1:def 9;
then y=(lower_sum_set(f)).D by INTEGRA1:def 11;
hence thesis by A7,A48,A47,A18;
end;
hence thesis;
end;
end;
hence thesis;
end;
theorem Th12:
i in dom D & j in dom D & i<=j & r < mid(D,i,j).1 implies
ex B be non empty closed_interval Subset of REAL st r = lower_bound B &
upper_bound B=mid(D,i,j).(len mid(D,i,j)) & mid(D,i,j) is Division of B
proof
assume
A1: i in dom D;
assume
A2: j in dom D;
assume i <= j;
then consider C being non empty closed_interval Subset of REAL such that
A3: lower_bound C=mid(D,i,j).1 and
A4: upper_bound C=mid(D,i,j).(len mid(D,i,j)) and
A5: mid(D,i,j) is Division of C by A1,A2,INTEGRA1:36;
reconsider MD=mid(D,i,j) as non empty increasing FinSequence of REAL by A5;
assume
A6: r < mid(D,i,j).1;
reconsider rr=r, ub=upper_bound C as Real;
ex a,b st a <= b & a = lower_bound C & b = upper_bound C by SEQ_4:11;
then r <= upper_bound C by A6,A3,XXREAL_0:2;
then reconsider B=[.rr,ub.]
as non empty closed_interval Subset of REAL by MEASURE5:14;
A7: B=[.lower_bound B,upper_bound B.] by INTEGRA1:4;
then
A8: lower_bound B = r by INTEGRA1:5;
A9: upper_bound B = upper_bound C by A7,INTEGRA1:5;
for x being Element of REAL holds x in C implies x in B
proof let x be Element of REAL;
assume
A10: x in C;
then lower_bound C <= x by INTEGRA2:1;
then
A11: r <= x by A6,A3,XXREAL_0:2;
x <= upper_bound C by A10,INTEGRA2:1;
hence thesis by A8,A9,A11,INTEGRA2:1;
end;
then
A12: C c= B;
rng mid(D,i,j) c= C by A5,INTEGRA1:def 2;
then rng mid(D,i,j) c= B by A12;
then MD is Division of B by A4,A9,INTEGRA1:def 2;
hence thesis by A4,A8,A9;
end;
Lm8: vol(A)<>0 & len D1 = 1 implies <*lower_bound A*>^D1 is non empty
increasing FinSequence of REAL
proof
assume
A1: vol(A)<>0;
reconsider lb = lower_bound A as Element of REAL by XREAL_0:def 1;
set MD1 = <*lb*>^D1;
A2: vol(A) >= 0 by INTEGRA1:9;
assume len D1 = 1;
then D1.1=upper_bound A by INTEGRA1:def 2;
then
A3: D1.1 - lower_bound A > 0 by A1,A2,INTEGRA1:def 5;
then
A4: lower_bound A < D1.1 by XREAL_1:47;
for n,m being Nat holds
n in dom MD1 & m in dom MD1 & n<m implies MD1.n < MD1.m
proof let n,m be Nat;
assume that
A5: n in dom MD1 and
A6: m in dom MD1 and
A7: n<m;
A8: not m in dom <*lower_bound A*>
proof
assume m in dom <*lower_bound A*>;
then m in Seg len <*lower_bound A*> by FINSEQ_1:def 3;
then m in {1} by FINSEQ_1:2,39;
then
A9: n < 1 by A7,TARSKI:def 1;
n in Seg len MD1 by A5,FINSEQ_1:def 3;
hence contradiction by A9,FINSEQ_1:1;
end;
A10: not MD1.m in rng <*lower_bound A*>
proof
assume MD1.m in rng <*lower_bound A*>;
then MD1.m in {lower_bound A} by FINSEQ_1:38;
then
A11: MD1.m = lower_bound A by TARSKI:def 1;
rng D1 <> {};
then
A12: 1 in dom D1 by FINSEQ_3:32;
consider n being Nat such that
A13: n in dom D1 and
A14: m=len <*lower_bound A*>+n by A6,A8,FINSEQ_1:25;
n in Seg len D1 by A13,FINSEQ_1:def 3;
then
A15: 1 <= n by FINSEQ_1:1;
D1.n=MD1.m by A13,A14,FINSEQ_1:def 7;
hence contradiction by A3,A11,A13,A15,A12,SEQ_4:137,XREAL_1:47;
end;
MD1.m in rng MD1 by A6,FUNCT_1:def 3;
then MD1.m in rng <*lower_bound A*> \/ rng D1 by FINSEQ_1:31;
then
A16: MD1.m in rng D1 by A10,XBOOLE_0:def 3;
now
per cases by A5,FINSEQ_1:25;
suppose
A17: n in dom <*lower_bound A*>;
then n in Seg len <*lower_bound A*> by FINSEQ_1:def 3;
then n in {1} by FINSEQ_1:2,39;
then
A18: n = 1 by TARSKI:def 1;
A19: MD1.n = <*lower_bound A*>.n by A17,FINSEQ_1:def 7
.= lower_bound A by A18;
rng D1 <> {};
then
A20: 1 in dom D1 by FINSEQ_3:32;
consider k such that
A21: k in dom D1 and
A22: MD1.m = D1.k by A16,PARTFUN1:3;
1 <= k by A21,FINSEQ_3:25;
then D1.1 <= MD1.m by A21,A22,A20,SEQ_4:137;
hence thesis by A4,A19,XXREAL_0:2;
end;
suppose
ex i be Nat st i in dom D1 & n = len <*lower_bound A*> + i;
then consider i such that
A23: i in dom D1 and
A24: n=len <*lower_bound A*> + i;
A25: D1.i=MD1.n by A23,A24,FINSEQ_1:def 7;
consider j be Nat such that
A26: j in dom D1 and
A27: m=len <*lower_bound A*> + j by A6,A8,FINSEQ_1:25;
A28: D1.j=MD1.m by A26,A27,FINSEQ_1:def 7;
i < j by A7,A24,A27,XREAL_1:6;
hence thesis by A23,A26,A25,A28,SEQM_3:def 1;
end;
end;
hence thesis;
end;
hence thesis by SEQM_3:def 1;
end;
Lm9: lower_bound A < D2.1 implies <*lower_bound A*>^D2 is non empty increasing
FinSequence of REAL
proof
reconsider lb = lower_bound A as Element of REAL by XREAL_0:def 1;
set MD2=<*lb*>^D2;
assume
A1: lower_bound A < D2.1;
for n,m being Nat holds
n in dom MD2 & m in dom MD2 & n<m implies MD2.n < MD2.m
proof let n,m be Nat;
assume that
A2: n in dom MD2 and
A3: m in dom MD2 and
A4: n<m;
A5: not m in dom <*lower_bound A*>
proof
assume m in dom <*lower_bound A*>;
then m in Seg len <*lower_bound A*> by FINSEQ_1:def 3;
then m in {1} by FINSEQ_1:2,39;
then
A6: n < 1 by A4,TARSKI:def 1;
n in Seg len MD2 by A2,FINSEQ_1:def 3;
hence contradiction by A6,FINSEQ_1:1;
end;
A7: not MD2.m in rng <*lower_bound A*>
proof
assume MD2.m in rng <*lower_bound A*>;
then MD2.m in {lower_bound A} by FINSEQ_1:38;
then
A8: MD2.m = lower_bound A by TARSKI:def 1;
rng D2 <> {};
then
A9: 1 in dom D2 by FINSEQ_3:32;
consider n be Nat such that
A10: n in dom D2 and
A11: m=len <*lower_bound A*>+n by A3,A5,FINSEQ_1:25;
n in Seg len D2 by A10,FINSEQ_1:def 3;
then
A12: 1 <= n by FINSEQ_1:1;
D2.n=MD2.m by A10,A11,FINSEQ_1:def 7;
hence contradiction by A1,A8,A10,A12,A9,SEQ_4:137;
end;
MD2.m in rng MD2 by A3,FUNCT_1:def 3;
then MD2.m in rng <*lower_bound A*> \/ rng D2 by FINSEQ_1:31;
then
A13: MD2.m in rng <*lower_bound A*> or MD2.m in rng D2 by XBOOLE_0:def 3;
now
per cases by A2,FINSEQ_1:25;
suppose
A14: n in dom <*lower_bound A*>;
then n in Seg len <*lower_bound A*> by FINSEQ_1:def 3;
then n in {1} by FINSEQ_1:2,39;
then
A15: n = 1 by TARSKI:def 1;
A16: MD2.n = <*lower_bound A*>.n by A14,FINSEQ_1:def 7
.= lower_bound A by A15;
rng D2 <> {};
then
A17: 1 in dom D2 by FINSEQ_3:32;
consider k such that
A18: k in dom D2 and
A19: MD2.m = D2.k by A13,A7,PARTFUN1:3;
k in Seg len D2 by A18,FINSEQ_1:def 3;
then 1 <= k by FINSEQ_1:1;
then D2.1 <= MD2.m by A18,A19,A17,SEQ_4:137;
hence thesis by A1,A16,XXREAL_0:2;
end;
suppose
ex i be Nat st i in dom D2 & n = len <*lower_bound A*> + i;
then consider i such that
A20: i in dom D2 and
A21: n=len <*lower_bound A*> + i;
A22: D2.i=MD2.n by A20,A21,FINSEQ_1:def 7;
consider j be Nat such that
A23: j in dom D2 and
A24: m=len <*lower_bound A*> + j by A3,A5,FINSEQ_1:25;
A25: D2.j=MD2.m by A23,A24,FINSEQ_1:def 7;
i < j by A4,A21,A24,XREAL_1:6;
hence thesis by A20,A23,A22,A25,SEQM_3:def 1;
end;
end;
hence thesis;
end;
hence thesis by SEQM_3:def 1;
end;
theorem Lm10:
for MD1 be Division of A holds MD1 = <*lower_bound A*>^D1 implies (for i
st i in Seg len D1 holds divset(MD1,i+1)=divset(D1,i)) & upper_volume(f,D1)=
upper_volume(f,MD1)/^1 & lower_volume(f,D1)=lower_volume(f,MD1)/^1
proof
let MD1 be Division of A;
assume
A1: MD1 = <*lower_bound A*>^D1;
thus
A2: for i st i in Seg len D1 holds divset(MD1,i+1)=divset(D1,i)
proof
let i;
assume
A3: i in Seg len D1;
then
A4: i in dom D1 by FINSEQ_1:def 3;
i <= len D1 by A3,FINSEQ_1:1;
then i+1 <= len D1+1 by XREAL_1:6;
then i+1 <= len D1 + len<*lower_bound A*> by FINSEQ_1:39;
then
A5: i+1 <= len MD1 by A1,FINSEQ_1:22;
1 <= i+1 by NAT_1:11;
then
A6: i+1 in dom MD1 by A5,FINSEQ_3:25;
A7: 1 <= i by A3,FINSEQ_1:1;
A8: lower_bound divset(D1,i)=lower_bound divset(MD1,i+1) & upper_bound
divset(D1,i)=upper_bound divset(MD1,i+1)
proof
per cases;
suppose
A9: i=1;
A10: i+1 > 1 by A7,NAT_1:13;
then lower_bound divset(MD1,i+1)=MD1.(i+1-1) by A6,INTEGRA1:def 4;
then
A11: lower_bound divset(MD1,i+1) = lower_bound A by A1,A9,FINSEQ_1:41;
A12: MD1.(i+1) = MD1.(i+len <*lower_bound A*>) by FINSEQ_1:40
.= D1.i by A1,A4,FINSEQ_1:def 7;
upper_bound divset(MD1,i+1)=MD1. (i+1) by A6,A10,INTEGRA1:def 4;
hence thesis by A4,A9,A11,A12,INTEGRA1:def 4;
end;
suppose
A13: i<>1;
A14: i+1 > 1 by A7,NAT_1:13;
MD1.(i+1) = MD1.(i+len <*lower_bound A*>) by FINSEQ_1:40
.= D1.i by A1,A4,FINSEQ_1:def 7;
then
A15: upper_bound divset(MD1,i+1)=D1.i by A6,A14,INTEGRA1:def 4
.= upper_bound divset(D1,i) by A4,A13,INTEGRA1:def 4;
i-1 in dom D1 by A4,A13,INTEGRA1:7;
then D1.(i-1) = MD1.(i-1+len <*lower_bound A*>) by A1,FINSEQ_1:def 7
.=MD1.(i-1+1) by FINSEQ_1:39
.=MD1.(i+1-1);
then lower_bound divset(D1,i)=MD1.(i+1-1) by A4,A13,INTEGRA1:def 4
.=lower_bound divset(MD1,i+1) by A6,A14,INTEGRA1:def 4;
hence thesis by A15;
end;
end;
divset(D1,i)=[.lower_bound divset(D1,i), upper_bound divset(D1,i).]
by INTEGRA1:4;
hence thesis by A8,INTEGRA1:4;
end;
A16: len MD1=len <*lower_bound A*> + len D1 by A1,FINSEQ_1:22
.= 1+len D1 by FINSEQ_1:39;
thus upper_volume(f,D1)=upper_volume(f,MD1)/^1
proof
set D2 = D1, MD2 = MD1;
rng upper_volume(f,MD2) <> {};
then 1 in dom upper_volume(f,MD2) by FINSEQ_3:32;
then 1 <= len upper_volume(f,MD2) by FINSEQ_3:25;
then len (upper_volume(f,MD2)/^1)=len upper_volume(f,MD2)-1 by
RFINSEQ:def 1
.=len MD2 -1 by INTEGRA1:def 6
.=len D2 by A16;
then
A17: len upper_volume(f,D2)=len (upper_volume(f,MD2)/^1) by INTEGRA1:def 6;
for k be Nat holds 1 <= k & k <= len upper_volume(f,D2) implies
upper_volume(f,D2).k = (upper_volume(f,MD2)/^1).k
proof
let k be Nat;
assume that
A18: 1 <= k and
A19: k <= len upper_volume(f,D2);
k+1 <= len upper_volume(f,D2)+1 by A19,XREAL_1:6;
then
A20: k+1 <= len D2+1 by INTEGRA1:def 6;
k in Seg len upper_volume(f,D2) by A18,A19,FINSEQ_1:1;
then
A21: k in Seg len D2 by INTEGRA1:def 6;
then k in dom D2 by FINSEQ_1:def 3;
then
A22: upper_volume(f,D2).k=(upper_bound rng(f|divset(D2,k)))*vol(divset(
D2,k )) by INTEGRA1:def 6
.=(upper_bound rng(f|divset(MD2,k+1)))*vol(divset(D2,k)) by A2,A21
.=(upper_bound rng(f|divset(MD2,k+1)))*vol(divset(MD2,k+1)) by A2,A21;
A23: len (upper_volume(f,MD2)/^1) <= len upper_volume(f,MD2) by FINSEQ_5:25;
1 <= k+1 by NAT_1:11;
then k+1 in Seg len MD2 by A16,A20,FINSEQ_1:1;
then
A24: k+1 in dom MD2 by FINSEQ_1:def 3;
1 <= len upper_volume(f,D2) by A18,A19,XXREAL_0:2;
then
A25: 1 <= len upper_volume(f,MD2) by A17,A23,XXREAL_0:2;
k in dom (upper_volume(f,MD2)/^1) by A17,A18,A19,FINSEQ_3:25;
then (upper_volume(f,MD2)/^1).k=upper_volume(f,MD2).(k+1) by A25,
RFINSEQ:def 1
.=(upper_bound rng(f|divset(MD2,k+1)))*vol(divset(MD2,k+1)) by A24,
INTEGRA1:def 6;
hence thesis by A22;
end;
hence thesis by A17,FINSEQ_1:14;
end;
rng lower_volume(f,MD1) <> {};
then 1 in dom lower_volume(f,MD1) by FINSEQ_3:32;
then 1 <= len lower_volume(f,MD1) by FINSEQ_3:25;
then len (lower_volume(f,MD1)/^1)=len lower_volume(f,MD1)-1 by RFINSEQ:def 1
.=len MD1 -1 by INTEGRA1:def 7
.=len D1 by A16;
then
A26: len lower_volume(f,D1)=len (lower_volume(f,MD1)/^1) by INTEGRA1:def 7;
for k be Nat holds 1 <= k & k <= len lower_volume(f,D1) implies
lower_volume(f,D1).k = (lower_volume(f,MD1)/^1).k
proof
let k be Nat;
assume that
A27: 1 <= k and
A28: k <= len lower_volume(f,D1);
A29: 1 <= k+1 by NAT_1:11;
k in Seg len lower_volume(f,D1) by A27,A28,FINSEQ_1:1;
then
A30: k in Seg len D1 by INTEGRA1:def 7;
then k in dom D1 by FINSEQ_1:def 3;
then
A31: lower_volume(f,D1).k=(lower_bound rng(f|divset(D1,k)))*vol(divset(D1,
k)) by INTEGRA1:def 7
.=(lower_bound rng(f|divset(MD1,k+1)))*vol(divset(D1,k)) by A2,A30
.=(lower_bound rng(f|divset(MD1,k+1)))*vol(divset(MD1,k+1)) by A2,A30;
A32: len (lower_volume(f,MD1)/^1) <= len lower_volume(f,MD1) by FINSEQ_5:25;
k+1 <= len lower_volume(f,D1)+1 by A28,XREAL_1:6;
then
A33: k+1 <= len D1+1 by INTEGRA1:def 7;
len MD1=len <*lower_bound A*>+len D1 by A1,FINSEQ_1:22
.=len D1 + 1 by FINSEQ_1:39;
then k+1 in Seg len MD1 by A29,A33,FINSEQ_1:1;
then
A34: k+1 in dom MD1 by FINSEQ_1:def 3;
1 <= len (lower_volume(f,MD1)/^1) by A26,A27,A28,XXREAL_0:2;
then
A35: 1 <= len lower_volume(f,MD1) by A32,XXREAL_0:2;
k in dom (lower_volume(f,MD1)/^1) by A26,A27,A28,FINSEQ_3:25;
then (lower_volume(f,MD1)/^1).k=lower_volume(f,MD1).(k+1) by A35,
RFINSEQ:def 1
.=(lower_bound rng(f|divset(MD1,k+1)))*vol(divset(MD1,k+1)) by A34,
INTEGRA1:def 7;
hence thesis by A31;
end;
hence thesis by A26,FINSEQ_1:14;
end;
Lm11: for MD2 being Division of A st MD2 = <*lower_bound A*>^D2 holds vol(
divset(MD2,1))=0
proof
let MD2 be Division of A;
assume
A1: MD2 = <*lower_bound A*>^D2;
rng MD2 <> {};
then
A2: 1 in dom MD2 by FINSEQ_3:32;
then
A3: upper_bound divset(MD2,1) = MD2.1 by INTEGRA1:def 4;
lower_bound divset(MD2,1) = lower_bound A by A2,INTEGRA1:def 4;
then vol(divset(MD2,1)) = MD2.1 - lower_bound A by A3,INTEGRA1:def 5
.=lower_bound A-lower_bound A by A1,FINSEQ_1:41;
hence thesis;
end;
Lm12: for MD1 be Division of A holds MD1 = <*lower_bound A*>^D1 implies delta(
MD1)=delta(D1)
proof
let MD1 be Division of A;
assume
A1: MD1 = <*lower_bound A*>^D1;
then
A2: vol(divset(MD1,1))=0 by Lm11;
delta(D1) in rng upper_volume(chi(A,A),D1) by XXREAL_2:def 8;
then consider i such that
A3: i in dom upper_volume(chi(A,A),D1) and
A4: upper_volume(chi(A,A),D1).i=delta(D1) by PARTFUN1:3;
delta(MD1) in rng upper_volume(chi(A,A),MD1) by XXREAL_2:def 8;
then consider j such that
A5: j in dom upper_volume(chi(A,A),MD1) and
A6: upper_volume(chi(A,A),MD1).j=delta(MD1 ) by PARTFUN1:3;
j in Seg len upper_volume(chi(A,A),MD1) by A5,FINSEQ_1:def 3;
then
A7: j in Seg len MD1 by INTEGRA1:def 6;
then
A8: j in dom MD1 by FINSEQ_1:def 3;
then
A9: delta(MD1)=(upper_bound rng(chi(A,A)|divset(MD1,j)))*vol(divset(MD1,j))
by A6,INTEGRA1:def 6;
A10: delta(MD1) <= delta(D1)
proof
per cases;
suppose
j=1;
hence thesis by A2,A9,Th9;
end;
suppose
j<>1;
then not j in Seg 1 by FINSEQ_1:2,TARSKI:def 1;
then not j in Seg len <*lower_bound A*> by FINSEQ_1:39;
then
A11: not j in dom <*lower_bound A*> by FINSEQ_1:def 3;
j in dom MD1 by A7,FINSEQ_1:def 3;
then consider k be Nat such that
A12: k in dom D1 and
A13: j=len <*lower_bound A*>+k by A1,A11,FINSEQ_1:25;
A14: k in Seg len D1 by A12,FINSEQ_1:def 3;
then divset(D1,k)=divset(MD1,k+1) by A1,Lm10
.=divset(MD1,j) by A13,FINSEQ_1:39;
then delta(MD1) = (upper_bound rng(chi(A,A)|divset(D1,k)))* vol(divset(
D1,k)) by A6,A8,INTEGRA1:def 6;
then
A15: delta(MD1)=upper_volume(chi(A,A),D1).k by A12,INTEGRA1:def 6;
k in Seg len upper_volume(chi(A,A),D1) by A14,INTEGRA1:def 6;
then k in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3;
then delta(MD1) in rng upper_volume(chi(A,A),D1) by A15,FUNCT_1:def 3;
hence thesis by XXREAL_2:def 8;
end;
end;
i in Seg len upper_volume(chi(A,A),D1) by A3,FINSEQ_1:def 3;
then
A16: i in Seg len D1 by INTEGRA1:def 6;
then i in dom D1 by FINSEQ_1:def 3;
then len <*lower_bound A*>+i in dom MD1 by A1,FINSEQ_1:28;
then
A17: i+1 in dom MD1 by FINSEQ_1:39;
then i+1 in Seg len MD1 by FINSEQ_1:def 3;
then i+1 in Seg len upper_volume(chi(A,A),MD1) by INTEGRA1:def 6;
then
A18: i+1 in dom upper_volume(chi(A,A),MD1) by FINSEQ_1:def 3;
i in dom D1 by A16,FINSEQ_1:def 3;
then delta(D1)=(upper_bound rng (chi(A,A)|divset(D1,i)))*vol(divset(D1,i))
by A4,INTEGRA1:def 6
.=(upper_bound rng(chi(A,A)|divset(MD1,i+1)))*vol(divset(D1,i)) by A1,A16
,Lm10
.=(upper_bound rng (chi(A,A)|divset(MD1,i+1)))*vol(divset(MD1,i+1)) by A1
,A16,Lm10;
then delta(D1)=upper_volume(chi(A,A),MD1).(i+1) by A17,INTEGRA1:def 6;
then delta(D1) in rng upper_volume(chi(A,A),MD1) by A18,FUNCT_1:def 3;
then delta(D1) <= delta(MD1) by XXREAL_2:def 8;
hence thesis by A10,XXREAL_0:1;
end;
theorem Th13:
x in divset(D1,len D1) & vol(A)<>0 & D1<=D2 & rng D2 = rng D1 \/
{x} & f|A is bounded & x > lower_bound A implies Sum lower_volume(f,D2)-Sum
lower_volume(f,D1)<=(upper_bound rng f-lower_bound rng f)*delta(D1)
proof
assume that
A1: x in divset(D1,len D1) and
A2: vol(A)<>0 and
A3: D1 <= D2 and
A4: rng D2 = rng D1 \/ {x} and
A5: f|A is bounded and
A6: x > lower_bound A;
len D1 in Seg len D1 by FINSEQ_1:3;
then
A7: 1 <= len D1 by FINSEQ_1:1;
then len D1 = 1 or len D1 > 1 by XXREAL_0:1;
then
A8: len D1 = 1 or len D1 >= 1+1 by NAT_1:13;
now
per cases by A8;
suppose
A9: len D1 = 1;
then reconsider MD1 = <*lower_bound A*>^D1 as non empty increasing
FinSequence of REAL by A2,Lm8;
A10: len MD1 = len <*lower_bound A*> + len D1 by FINSEQ_1:22;
len <*lower_bound A*> + 1 <= len <*lower_bound A*> + len D1 by A7,
XREAL_1:6;
then
MD1.(len MD1)=D1.(len <*lower_bound A*>+ len D1-len <*lower_bound A
*>) by A10,FINSEQ_1:23
.=D1.(len D1);
then
A11: MD1.(len MD1) = upper_bound A by INTEGRA1:def 2;
for y being Element of REAL holds y in rng MD1 implies y in A
proof let y be Element of REAL;
assume y in rng MD1;
then
A12: y in rng <*lower_bound A*> \/ rng D1 by FINSEQ_1:31;
per cases by A12,XBOOLE_0:def 3;
suppose
y in rng <*lower_bound A*>;
then y in {lower_bound A} by FINSEQ_1:38;
then
A13: y = lower_bound A by TARSKI:def 1;
ex a,b st a <= b & a = lower_bound A & b = upper_bound A by SEQ_4:11;
hence thesis by A13,INTEGRA2:1;
end;
suppose
A14: y in rng D1;
rng D1 c= A by INTEGRA1:def 2;
hence thesis by A14;
end;
end;
then rng MD1 c= A;
then reconsider MD1 as Division of A by A11,INTEGRA1:def 2;
A15: len MD1=len <*lower_bound A*> + len D1 by FINSEQ_1:22
.= 1+len D1 by FINSEQ_1:39;
A16: vol(A) >= 0 by INTEGRA1:9;
D1.1=upper_bound A by A9,INTEGRA1:def 2;
then D1.1 - lower_bound A > 0 by A2,A16,INTEGRA1:def 5;
then
A17: lower_bound A < D1.1 by XREAL_1:47;
lower_volume(f,D1)=lower_volume(f,MD1)/^1 by Lm10;
then
lower_volume(f,MD1)=<*(lower_volume(f,MD1))/.1*>^lower_volume(f,D1)
by FINSEQ_5:29;
then
A18: Sum lower_volume(f,MD1)=((lower_volume(f,MD1))/.1)+Sum lower_volume
(f, D1) by RVSUM_1:76;
A19: len D1 in dom D1 by FINSEQ_5:6;
A20: 1+len D1 >= 1+1 by A7,XREAL_1:6;
then
A21: len MD1 <> 1 by A15;
A22: len MD1 in dom MD1 by FINSEQ_5:6;
len MD1 - 1 =len D1 by A15;
then lower_bound divset(MD1,len MD1) = MD1.(len D1) by A22,A21,
INTEGRA1:def 4
.=lower_bound A by A9,FINSEQ_1:41;
then
A23: lower_bound divset(D1,len D1) = lower_bound divset(MD1,len MD1) by A9,A19
,INTEGRA1:def 4;
set MD2=<*lower_bound A*>^D2;
rng MD1 <> {};
then
A24: 1 in dom MD1 by FINSEQ_3:32;
then
A25: lower_volume(f,MD1).1 = (lower_bound rng(f|divset(MD1,1)))* vol(
divset( MD1,1)) by INTEGRA1:def 7;
1 in Seg len MD1 by A24,FINSEQ_1:def 3;
then 1 in Seg len lower_volume(f,MD1) by INTEGRA1:def 7;
then
A26: 1 in dom lower_volume(f,MD1) by FINSEQ_1:def 3;
rng D2 <> {};
then
A27: 1 in dom D2 by FINSEQ_3:32;
then 1 <= len D2 by FINSEQ_3:25;
then
A28: len <*lower_bound A*> + 1 <= len <*lower_bound A*> + len D2 by XREAL_1:6;
A29: D2.1 in rng D2 by A27,FUNCT_1:def 3;
lower_bound A < D2.1
proof
per cases by A4,A29,XBOOLE_0:def 3;
suppose
A30: D2.1 in rng D1;
rng D1 <> {};
then
A31: 1 in dom D1 by FINSEQ_3:32;
consider k such that
A32: k in dom D1 and
A33: D1.k = D2.1 by A30,PARTFUN1:3;
1 <= k by A32,FINSEQ_3:25;
then D1.1 <= D2.1 by A32,A33,A31,SEQ_4:137;
hence thesis by A17,XXREAL_0:2;
end;
suppose
D2.1 in {x};
hence thesis by A6,TARSKI:def 1;
end;
end;
then reconsider MD2 as non empty increasing FinSequence of REAL by Lm9;
len MD2 = len <*lower_bound A*> + len D2 by FINSEQ_1:22;
then
MD2.(len MD2)=D2.(len <*lower_bound A*>+len D2- len <*lower_bound A
*>) by A28,FINSEQ_1:23
.=D2.(len D2);
then
A34: MD2.(len MD2) = upper_bound A by INTEGRA1:def 2;
for y being Element of REAL holds y in rng MD2 implies y in A
proof let y be Element of REAL;
assume y in rng MD2;
then
A35: y in rng <*lower_bound A*> \/ rng D2 by FINSEQ_1:31;
per cases by A35,XBOOLE_0:def 3;
suppose
y in rng <*lower_bound A*>;
then y in {lower_bound A} by FINSEQ_1:38;
then
A36: y = lower_bound A by TARSKI:def 1;
ex a,b st a <= b & a = lower_bound A & b = upper_bound A by SEQ_4:11;
hence thesis by A36,INTEGRA2:1;
end;
suppose
A37: y in rng D2;
rng D2 c= A by INTEGRA1:def 2;
hence thesis by A37;
end;
end;
then rng MD2 c= A;
then reconsider MD2 as Division of A by A34,INTEGRA1:def 2;
A38: x <= upper_bound divset(D1,len D1) by A1,INTEGRA2:1;
rng MD2 = rng D2 \/ rng <*lower_bound A*> by FINSEQ_1:31
.=rng D1 \/ rng <*lower_bound A*> \/ {x} by A4,XBOOLE_1:4;
then
A39: rng MD2 = rng MD1 \/ {x} by FINSEQ_1:31;
MD1.(len MD1) = MD1.(len <*lower_bound A*> + len D1) by FINSEQ_1:22
.=D1.(len D1) by A19,FINSEQ_1:def 7;
then
A40: upper_bound divset(MD1,len MD1)=D1.(len D1) by A22,A21,INTEGRA1:def 4
.= upper_bound divset(D1,len D1) by A9,A19,INTEGRA1:def 4;
rng D1 c= rng D2 by A3,INTEGRA1:def 18;
then rng D1 \/ rng<*lower_bound A*> c= rng D2 \/ rng <*lower_bound A*>
by XBOOLE_1:9;
then rng MD1 c= rng D2 \/ rng <*lower_bound A*> by FINSEQ_1:31;
then
A41: rng MD1 c= rng MD2 by FINSEQ_1:31;
len D1 <= len D2 by A3,INTEGRA1:def 18;
then
len D1+len <*lower_bound A*> <= len D2+len <*lower_bound A*> by XREAL_1:6;
then len MD1 <= len D2+len <*lower_bound A*> by FINSEQ_1:22;
then len MD1 <= len MD2 by FINSEQ_1:22;
then
A42: MD1<=MD2 by A41,INTEGRA1:def 18;
lower_bound divset(D1,len D1) <= x by A1,INTEGRA2:1;
then x in divset(MD1,len MD1) by A38,A23,A40,INTEGRA2:1;
then
A43: Sum lower_volume(f,MD2)-Sum lower_volume(f,MD1)<=(upper_bound rng f
-lower_bound rng f)*delta(MD1) by A5,A15,A20,A42,A39,Th10;
rng MD2 <> {};
then
A44: 1 in dom MD2 by FINSEQ_3:32;
then
A45: lower_volume(f,MD2).1 = (lower_bound rng(f|divset(MD2,1)))* vol(
divset( MD2,1)) by INTEGRA1:def 7;
1 in Seg len MD2 by A44,FINSEQ_1:def 3;
then 1 in Seg len lower_volume(f,MD2) by INTEGRA1:def 7;
then
A46: 1 in dom lower_volume(f,MD2) by FINSEQ_1:def 3;
vol(divset(MD2,1))=0 by Lm11;
then
A47: lower_volume(f,MD2)/.1 = 0 by A45,A46,PARTFUN1:def 6;
lower_volume(f,D2)=lower_volume(f,MD2)/^1 by Lm10;
then
lower_volume(f,MD2)=<*(lower_volume(f,MD2))/.1*>^lower_volume(f,D2)
by FINSEQ_5:29;
then
A48: Sum lower_volume(f,MD2)=(lower_volume(f,MD2))/.1+Sum lower_volume(f
,D2 ) by RVSUM_1:76;
vol(divset(MD1,1))=0 by Lm11;
then lower_volume(f,MD1)/.1 = 0 by A25,A26,PARTFUN1:def 6;
hence thesis by A43,A18,A48,A47,Lm12;
end;
suppose
len D1 >= 2;
hence thesis by A1,A3,A4,A5,Th10;
end;
end;
hence thesis;
end;
theorem Th14:
x in divset(D1,len D1) & vol(A)<>0 & D1<=D2 & rng D2 = rng D1 \/
{x} & f|A is bounded & x > lower_bound A implies Sum upper_volume(f,D1)-Sum
upper_volume(f,D2)<=(upper_bound rng f-lower_bound rng f)*delta(D1)
proof
assume that
A1: x in divset(D1,len D1) and
A2: vol(A)<>0 and
A3: D1 <= D2 and
A4: rng D2 = rng D1 \/ {x} and
A5: f|A is bounded and
A6: x > lower_bound A;
len D1 in Seg len D1 by FINSEQ_1:3;
then
A7: 1 <= len D1 by FINSEQ_1:1;
then len D1 = 1 or len D1 > 1 by XXREAL_0:1;
then
A8: len D1 = 1 or len D1 >= 1+1 by NAT_1:13;
now
per cases by A8;
suppose
A9: len D1 = 1;
then reconsider MD1 = <*lower_bound A*>^D1 as non empty increasing
FinSequence of REAL by A2,Lm8;
A10: len MD1 = len <*lower_bound A*> + len D1 by FINSEQ_1:22;
len <*lower_bound A*> + 1 <= len <*lower_bound A*> + len D1 by A7,
XREAL_1:6;
then
MD1.(len MD1)=D1.(len <*lower_bound A*>+len D1- len <*lower_bound A
*>) by A10,FINSEQ_1:23
.=D1.(len D1);
then
A11: MD1.(len MD1) = upper_bound A by INTEGRA1:def 2;
for y being Element of REAL holds y in rng MD1 implies y in A
proof let y be Element of REAL;
assume y in rng MD1;
then
A12: y in rng <*lower_bound A*> \/ rng D1 by FINSEQ_1:31;
per cases by A12,XBOOLE_0:def 3;
suppose
y in rng <*lower_bound A*>;
then y in {lower_bound A} by FINSEQ_1:38;
then
A13: y = lower_bound A by TARSKI:def 1;
ex a,b st a <= b & a = lower_bound A & b = upper_bound A by SEQ_4:11;
hence thesis by A13,INTEGRA2:1;
end;
suppose
A14: y in rng D1;
rng D1 c= A by INTEGRA1:def 2;
hence thesis by A14;
end;
end;
then rng MD1 c= A;
then reconsider MD1 as Division of A by A11,INTEGRA1:def 2;
A15: len MD1=len <*lower_bound A*> + len D1 by FINSEQ_1:22
.= 1+len D1 by FINSEQ_1:39;
A16: vol(A) >= 0 by INTEGRA1:9;
D1.1=upper_bound A by A9,INTEGRA1:def 2;
then D1.1 - lower_bound A > 0 by A2,A16,INTEGRA1:def 5;
then
A17: lower_bound A < D1.1 by XREAL_1:47;
upper_volume(f,D1)=upper_volume(f,MD1)/^1 by Lm10;
then upper_volume(f,MD1)=<*upper_volume(f,MD1)/.1*>^upper_volume(f,D1)
by FINSEQ_5:29;
then
A18: Sum upper_volume(f,MD1)=(upper_volume(f,MD1)/.1)+Sum upper_volume(f
,D1 ) by RVSUM_1:76;
A19: len D1 in dom D1 by FINSEQ_5:6;
A20: 1+len D1 >= 1+1 by A7,XREAL_1:6;
then
A21: len MD1 <> 1 by A15;
A22: len MD1 in dom MD1 by FINSEQ_5:6;
len MD1 - 1 = len D1 by A15;
then lower_bound divset(MD1,len MD1) = MD1.(len D1) by A22,A21,
INTEGRA1:def 4
.=lower_bound A by A9,FINSEQ_1:41;
then
A23: lower_bound divset(D1,len D1) = lower_bound divset(MD1,len MD1) by A9,A19
,INTEGRA1:def 4;
set MD2=<*lower_bound A*>^D2;
rng MD1 <> {};
then
A24: 1 in dom MD1 by FINSEQ_3:32;
then
A25: upper_volume(f,MD1).1 = (upper_bound rng(f|divset(MD1,1)))* vol(
divset( MD1,1)) by INTEGRA1:def 6;
1 in Seg len MD1 by A24,FINSEQ_1:def 3;
then 1 in Seg len upper_volume(f,MD1) by INTEGRA1:def 6;
then
A26: 1 in dom upper_volume(f,MD1) by FINSEQ_1:def 3;
rng D2 <> {};
then
A27: 1 in dom D2 by FINSEQ_3:32;
then 1 <= len D2 by FINSEQ_3:25;
then
A28: len <*lower_bound A*> + 1 <= len <*lower_bound A*> + len D2 by XREAL_1:6;
A29: D2.1 in rng D2 by A27,FUNCT_1:def 3;
lower_bound A < D2.1
proof
per cases by A4,A29,XBOOLE_0:def 3;
suppose
A30: D2.1 in rng D1;
rng D1 <> {};
then
A31: 1 in dom D1 by FINSEQ_3:32;
consider k such that
A32: k in dom D1 and
A33: D1.k = D2.1 by A30,PARTFUN1:3;
1 <= k by A32,FINSEQ_3:25;
then D1.1 <= D2.1 by A32,A33,A31,SEQ_4:137;
hence thesis by A17,XXREAL_0:2;
end;
suppose
D2.1 in {x};
hence thesis by A6,TARSKI:def 1;
end;
end;
then reconsider MD2 as non empty increasing FinSequence of REAL by Lm9;
len MD2 = len <*lower_bound A*> + len D2 by FINSEQ_1:22;
then
MD2.(len MD2)=D2.(len <*lower_bound A*>+len D2- len <*lower_bound A
*>) by A28,FINSEQ_1:23
.=D2.(len D2);
then
A34: MD2.(len MD2) = upper_bound A by INTEGRA1:def 2;
for y being Element of REAL holds y in rng MD2 implies y in A
proof let y be Element of REAL;
assume y in rng MD2;
then
A35: y in rng <*lower_bound A*> \/ rng D2 by FINSEQ_1:31;
per cases by A35,XBOOLE_0:def 3;
suppose
y in rng <*lower_bound A*>;
then y in {lower_bound A} by FINSEQ_1:38;
then
A36: y = lower_bound A by TARSKI:def 1;
ex a,b st a <= b & a = lower_bound A & b = upper_bound A by SEQ_4:11;
hence thesis by A36,INTEGRA2:1;
end;
suppose
A37: y in rng D2;
rng D2 c= A by INTEGRA1:def 2;
hence thesis by A37;
end;
end;
then rng MD2 c= A;
then reconsider MD2 as Division of A by A34,INTEGRA1:def 2;
A38: x <= upper_bound divset(D1,len D1) by A1,INTEGRA2:1;
rng MD2 = rng D2 \/ rng <*lower_bound A*> by FINSEQ_1:31
.=rng D1 \/ rng <*lower_bound A*> \/ {x} by A4,XBOOLE_1:4;
then
A39: rng MD2 = rng MD1 \/ {x} by FINSEQ_1:31;
MD1.(len MD1) = MD1.(len <*lower_bound A*> + len D1) by FINSEQ_1:22
.=D1.(len D1) by A19,FINSEQ_1:def 7;
then
A40: upper_bound divset(MD1,len MD1)=D1.(len D1) by A22,A21,INTEGRA1:def 4
.= upper_bound divset(D1,len D1) by A9,A19,INTEGRA1:def 4;
rng D1 c= rng D2 by A3,INTEGRA1:def 18;
then rng D1 \/ rng<*lower_bound A*> c= rng D2 \/ rng <*lower_bound A*>
by XBOOLE_1:9;
then rng MD1 c= rng D2 \/ rng <*lower_bound A*> by FINSEQ_1:31;
then
A41: rng MD1 c= rng MD2 by FINSEQ_1:31;
len D1 <= len D2 by A3,INTEGRA1:def 18;
then
len D1+len <*lower_bound A*> <= len D2+len <*lower_bound A*> by XREAL_1:6;
then len MD1 <= len D2+len <*lower_bound A*> by FINSEQ_1:22;
then len MD1 <= len MD2 by FINSEQ_1:22;
then
A42: MD1<=MD2 by A41,INTEGRA1:def 18;
lower_bound divset(D1,len D1) <= x by A1,INTEGRA2:1;
then x in divset(MD1,len MD1) by A38,A23,A40,INTEGRA2:1;
then
A43: Sum upper_volume(f,MD1)-Sum upper_volume(f,MD2) <=(upper_bound rng
f-lower_bound rng f)*delta(MD1) by A5,A15,A20,A42,A39,Th11;
rng MD2 <> {};
then
A44: 1 in dom MD2 by FINSEQ_3:32;
then
A45: upper_volume(f,MD2).1 = (upper_bound rng(f|divset(MD2,1)))* vol(
divset( MD2,1)) by INTEGRA1:def 6;
1 in Seg len MD2 by A44,FINSEQ_1:def 3;
then 1 in Seg len upper_volume(f,MD2) by INTEGRA1:def 6;
then
A46: 1 in dom upper_volume(f,MD2) by FINSEQ_1:def 3;
vol(divset(MD2,1))=0 by Lm11;
then
A47: upper_volume(f,MD2)/.1 = 0 by A45,A46,PARTFUN1:def 6;
upper_volume(f,D2)=upper_volume(f,MD2)/^1 by Lm10;
then upper_volume(f,MD2)=<*upper_volume(f,MD2)/.1*>^upper_volume(f,D2)
by FINSEQ_5:29;
then
A48: Sum upper_volume(f,MD2)=(upper_volume(f,MD2)/.1)+Sum upper_volume(f
,D2 ) by RVSUM_1:76;
vol(divset(MD1,1))=0 by Lm11;
then upper_volume(f,MD1)/.1 = 0 by A25,A26,PARTFUN1:def 6;
hence thesis by A43,A18,A48,A47,Lm12;
end;
suppose
len D1 >= 2;
hence thesis by A1,A3,A4,A5,Th11;
end;
end;
hence thesis;
end;
theorem Th15:
i in dom D1 & j in dom D1 & i<=j & D1 <= D2 & r < mid(D2,indx(D2
,D1,i),indx(D2,D1,j)).1 implies
ex B be non empty closed_interval Subset of REAL, MD1,MD2
be Division of B st r=lower_bound B & upper_bound B=MD2.(len MD2) & upper_bound
B=MD1.(len MD1) & MD1 <= MD2 & MD1=mid(D1,i,j) & MD2=mid(D2,indx(D2,D1,i),indx(
D2,D1,j))
proof
set MD1=mid(D1,i,j);
set MD2=mid(D2,indx(D2,D1,i),indx(D2,D1,j));
assume
A1: i in dom D1;
then
A2: 1 <= i by FINSEQ_3:25;
assume
A3: j in dom D1;
assume
A4: i <= j;
then j-i >= 0 by XREAL_1:48;
then
A5: j-i+1 >= 0+1 by XREAL_1:6;
A6: j <= len D1 by A3,FINSEQ_3:25;
then
A7: MD1.1 = D1.(1+i-1) by A4,A5,A2,FINSEQ_6:122
.= D1.i;
assume
A8: D1 <= D2;
then
A9: D2.indx(D2,D1,i)=D1.i by A1,INTEGRA1:def 19;
A10: D2.indx(D2,D1,j)=D1.j by A3,A8,INTEGRA1:def 19;
A11: indx(D2,D1,i) in dom D2 by A1,A8,INTEGRA1:def 19;
then
A12: 1 <= indx(D2,D1,i) by FINSEQ_3:25;
A13: indx(D2,D1,j) in dom D2 by A3,A8,INTEGRA1:def 19;
then
A14: indx(D2,D1,j) <= len D2 by FINSEQ_3:25;
D1.i <= D1.j by A1,A3,A4,SEQ_4:137;
then
A15: indx(D2,D1,i) <= indx(D2,D1,j) by A11,A9,A13,A10,SEQM_3:def 1;
assume
A16: r < mid(D2,indx(D2,D1,i),indx(D2,D1,j)).1;
then consider B being non empty closed_interval Subset of REAL such that
A17: r = lower_bound B and
A18: upper_bound B=MD2.(len MD2) and
A19: MD2 is Division of B by A11,A13,A15,Th12;
A20: len MD2=indx(D2,D1,j)-indx(D2,D1,i)+1 by A11,A13,A15,INTEGRA1:58;
reconsider MD2 as Division of B by A19;
indx(D2,D1,j)-indx(D2,D1,i) >= 0 by A15,XREAL_1:48;
then
A21: indx(D2,D1,j)-indx(D2,D1,i)+1 >= 0+1 by XREAL_1:6;
then
A22: MD2.(len MD2)=D2.(indx(D2,D1,j)-indx(D2,D1,i)+1-1+indx(D2,D1,i)) by A15
,A20,A12,A14,FINSEQ_6:122
.= D1.j by A3,A8,INTEGRA1:def 19;
MD2.1=D2.(1+indx(D2,D1,i)-1) by A15,A21,A12,A14,FINSEQ_6:122
.=D1.i by A1,A8,INTEGRA1:def 19;
then consider C being non empty closed_interval Subset of REAL such that
A23: r = lower_bound C and
A24: upper_bound C=MD1.(len MD1) and
A25: MD1 is Division of C by A1,A3,A4,A16,A7,Th12;
len MD1=j-i+1 by A1,A3,A4,INTEGRA1:58;
then A26: MD1.(len MD1)=D1.(j-i+1-1+i) by A4,A5,A2,A6,FINSEQ_6:122
.=D1.j;
A27: B=[.lower_bound B,upper_bound B.] by INTEGRA1:4
.=C by A17,A18,A23,A24,A26,A22,INTEGRA1:4;
then reconsider MD1 as Division of B by A25;
A28: rng MD1 c= rng MD2
proof
let x1 be object;
A29: rng MD1 c= rng D1 by FINSEQ_6:119;
assume
A30: x1 in rng MD1;
then consider k1 being Element of NAT such that
A31: k1 in dom MD1 and
A32: MD1.k1=x1 by PARTFUN1:3;
rng D1 c= rng D2 by A8,INTEGRA1:def 18;
then rng MD1 c= rng D2 by A29;
then consider k2 being Element of NAT such that
A33: k2 in dom D2 and
A34: D2.k2=x1 by A30,PARTFUN1:3;
A35: k1 <= len MD1 by A31,FINSEQ_3:25;
A36: 1 <= k1 by A31,FINSEQ_3:25;
then 1 <= len MD1 by A35,XXREAL_0:2;
then 1 in dom MD1 by FINSEQ_3:25;
then MD1.1<=MD1.k1 by A31,A36,SEQ_4:137;
then
A37: indx(D2,D1,i) <= k2 by A11,A9,A7,A33,A34,A32,SEQM_3:def 1;
then consider k3 being Nat such that
A38: k2+1 = indx(D2,D1,i)+k3 by NAT_1:10,12;
len MD1 in dom MD1 by FINSEQ_5:6;
then MD1.k1<=MD1.(len MD1) by A31,A35,SEQ_4:137;
then k2 <= indx(D2,D1,j) by A13,A10,A26,A33,A34,A32,SEQM_3:def 1;
then k2+1<=indx(D2,D1,j)+1 by XREAL_1:6;
then
A39: k2+1-indx(D2,D1,i)<=indx(D2,D1,j)+1-indx( D2, D1,i) by XREAL_1:9;
indx(D2,D1,i)+1<=k2+1 by A37,XREAL_1:6;
then
A40: 1<=k2+1-indx(D2,D1,i) by XREAL_1:19;
then
A41: k3 in dom MD2 by A20,A39,A38,FINSEQ_3:25;
MD2.k3 = D2.(k3+indx(D2,D1,i)-1) by A15,A12,A14,A40,A39,A38,
FINSEQ_6:122;
hence thesis by A34,A38,A41,FUNCT_1:def 3;
end;
A42: card(rng MD2) = len MD2 by FINSEQ_4:62;
card(rng MD1) = len MD1 by FINSEQ_4:62;
then len MD1 <= len MD2 by A28,A42,NAT_1:43;
then MD1 <= MD2 by A28,INTEGRA1:def 18;
hence thesis by A17,A18,A24,A27;
end;
theorem Th16:
x in rng D implies D.1 <= x & x <= D.(len D)
proof
assume x in rng D;
then consider i such that
A1: i in dom D and
A2: x=D.i by PARTFUN1:3;
A3: i <= len D by A1,FINSEQ_3:25;
A4: 1 <= i by A1,FINSEQ_3:25;
then
A5: 1 <= len D by A3,XXREAL_0:2;
then
A6: len D in dom D by FINSEQ_3:25;
1 in dom D by A5,FINSEQ_3:25;
hence thesis by A1,A2,A4,A3,A6,SEQ_4:137;
end;
theorem Th17:
for p be FinSequence of REAL, i,j,k st p is increasing & i in
dom p & j in dom p & k in dom p & p.i <= p.k & p.k <= p.j holds p.k in rng mid(
p,i,j)
proof
let p be FinSequence of REAL;
let i,j,k;
assume that
A1: p is increasing and
A2: i in dom p and
A3: j in dom p and
A4: k in dom p and
A5: p.i <= p.k and
A6: p.k <= p.j;
A7: 1 <= i by A2,FINSEQ_3:25;
A8: 1 <= j by A3,FINSEQ_3:25;
A9: j <= len p by A3,FINSEQ_3:25;
A10: i <= k by A1,A2,A4,A5,SEQM_3:def 1;
then consider n being Nat such that
A11: k+1=i+n by NAT_1:10,12;
A12: k <= j by A1,A3,A4,A6,SEQM_3:def 1;
then k-i <= j-i by XREAL_1:9;
then
A13: k-i+1 <= j-i+1 by XREAL_1:6;
k-i >= 0 by A10,XREAL_1:48;
then
A14: k-i+1 >= 0+1 by XREAL_1:6;
A15: i <= j by A10,A12,XXREAL_0:2;
i <= len p by A2,FINSEQ_3:25;
then len mid(p,i,j) = j-'i+1 by A7,A8,A9,A15,FINSEQ_6:118;
then len mid(p,i,j) = j-i+1 by A10,A12,XREAL_1:233,XXREAL_0:2;
then
A16: n in dom mid(p,i,j) by A11,A14,A13,FINSEQ_3:25;
mid(p,i,j).n = p.(n+i-1) by A7,A9,A15,A11,A14,A13,FINSEQ_6:122
.=p.k by A11;
hence thesis by A16,FUNCT_1:def 3;
end;
theorem Th18:
f|A is bounded & i in dom D implies lower_bound rng(f|divset(D,i
)) <= upper_bound rng f
proof
assume
A1: f|A is bounded;
assume i in dom D;
then divset(D,i) c= A by INTEGRA1:8;
hence thesis by A1,Lm4;
end;
theorem Th19:
f|A is bounded & i in dom D implies upper_bound rng(f|divset(D,i
)) >= lower_bound rng f
proof
assume
A1: f|A is bounded;
assume i in dom D;
then divset(D,i) c= A by INTEGRA1:8;
hence thesis by A1,Lm4;
end;
begin :: Darboux's Theorem
theorem Th20:
f|A is bounded implies
for D,D1 ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D &
0 <= lower_sum(f,D2)-lower_sum(f,D) & 0<=lower_sum(f,D2)-lower_sum(f,D1)
proof
assume
A1: f|A is bounded;
for D,D1 ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D &
0 <= lower_sum(f,D2)-lower_sum(f,D) & 0<=lower_sum(f,D2)-lower_sum(f,D1)
proof
let D,D1;
consider D2 such that
A6: D<=D2 and
A7: D1<=D2 and
A8: rng D2=rng D1 \/ rng D by Th4;
A9: lower_sum(f,D2)-lower_sum(f,D1)>=0 by A1,A7,INTEGRA1:46,XREAL_1:48;
lower_sum(f,D2)-lower_sum(f,D)>=0 by A1,A6,INTEGRA1:46,XREAL_1:48;
hence thesis by A6,A7,A8,A9;
end;
hence thesis;
end;
theorem Th21:
f|A is bounded implies
for D,D1 st delta(D1)<min rng upper_volume(chi(A,A),D) ex D2 st D<=D2 &
D1<=D2 & rng D2=rng D1 \/ rng D & lower_sum(f,D2)-lower_sum(f,D1) <=
(len D)*(upper_bound(rng f)-lower_bound(rng f))*delta(D1)
proof
assume
A1: f|A is bounded;
then
A5: for D,D1 ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D &
0 <= lower_sum(f,D2)-lower_sum(f,D) & 0<=lower_sum(f,D2)-lower_sum(f,D1)
by Th20;
for D,D1 st delta(D1)<min rng upper_volume(chi(A,A),D) ex D2 st D<=D2 &
D1<=D2 & rng D2=rng D1 \/ rng D & lower_sum(f,D2)-lower_sum(f,D1) <= (len D)*(
upper_bound(rng f)-lower_bound(rng f))*delta(D1)
proof
let D,D1;
assume
A11: delta(D1)<min rng upper_volume(chi(A,A),D);
ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & lower_sum(f,D2)-
lower_sum(f,D1)<=(len D)*(upper_bound(rng f)- lower_bound(rng f))*delta(D1)
proof
consider D2 such that
A12: D<=D2 and
A13: D1<=D2 and
A14: rng D2=rng D1 \/ rng D and
0<=lower_sum(f,D2)-lower_sum(f,D) and
0<=lower_sum(f,D2)-lower_sum(f,D1) by A5;
lower_sum(f,D2)-lower_sum(f,D1) <= (len D)*(upper_bound(rng f)-
lower_bound(rng f))*delta(D1)
proof
deffunc LVf(Division of A) = lower_volume(f,$1);
deffunc PLf(Division of A,Nat) = (PartSums(lower_volume(f,$1))).$2;
A15: len D2 in dom D2 by FINSEQ_5:6;
A16: for i st i in dom D holds ex j st j in dom D1 & D.i in divset(D1,
j) & PLf(D2,indx(D2,D1,j))-PLf(D1,j)<=i*(upper_bound(rng f)- lower_bound(rng f)
)*delta(D1)
proof
defpred P[non zero Nat] means $1 in dom D implies ex j st j in dom
D1 & D.$1 in divset(D1,j) & PLf(D2,indx(D2,D1,j))-PLf(D1,j)<= $1*(upper_bound
rng f-lower_bound rng f)*delta(D1);
let i;
assume
A17: i in dom D;
then
A18: i in Seg len D by FINSEQ_1:def 3;
A19: for i,j st i in dom D & j in dom D1 & D.i in divset(D1,j) holds
j >= 2
proof
let i,j;
assume
A20: i in dom D;
assume that
A21: j in dom D1 and
A22: D.i in divset(D1,j);
assume j<2;
then j<1+1;
then
A23: j <= 1 by NAT_1:13;
j in Seg len D1 by A21,FINSEQ_1:def 3;
then j >= 1 by FINSEQ_1:1;
then j = 1 by A23,XXREAL_0:1;
then
A24: lower_bound divset(D1,j)=lower_bound A by A21,INTEGRA1:def 4;
A25: D.i<=upper_bound divset(D1,j) by A22,INTEGRA2:1;
delta(D1) >= min rng upper_volume(chi(A,A),D)
proof
per cases;
suppose
A26: i=1;
len D in Seg len D by FINSEQ_1:3;
then 1 <= len D by FINSEQ_1:1;
then
A27: 1 in Seg len D by FINSEQ_1:1;
then
A28: 1 in dom D by FINSEQ_1:def 3;
then
A29: lower_bound divset(D,1)=lower_bound A by INTEGRA1:def 4;
1 in Seg len upper_volume(chi(A,A),D) by A27,INTEGRA1:def 6;
then
A30: 1 in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3;
vol(divset(D,1)) = upper_volume(chi(A,A),D).1 by A28,
INTEGRA1:20;
then vol(divset(D,1)) in rng upper_volume(chi(A,A),D) by A30,
FUNCT_1:def 3;
then
A31: vol
(divset(D,1))>=min rng upper_volume(chi(A,A),D) by XXREAL_2:def 7;
A32: upper_bound divset(D,1)=D. 1 by A28,INTEGRA1:def 4;
upper_bound divset(D1,j)-lower_bound A >= D.1-lower_bound
A by A25,A26,XREAL_1:9;
then vol(divset(D1,j)) >= upper_bound divset(D,1)-lower_bound
divset(D,1) by A24,A29,A32,INTEGRA1:def 5;
then
A33: vol(divset(D1,j)) >= vol(divset(D,1)) by INTEGRA1:def 5;
vol(divset(D1,j)) <= delta(D1) by A21,Lm5;
then delta(D1) >= vol(divset(D,1)) by A33,XXREAL_0:2;
hence thesis by A31,XXREAL_0:2;
end;
suppose
A34: i<>1;
then D.(i-1) in A by A20,INTEGRA1:7;
then
A35: lower_bound A <= D.(i-1) by INTEGRA2:1;
lower_bound divset(D,i)=D.(i-1) by A20,A34,INTEGRA1:def 4;
then
A36: upper_bound divset(D,i)-lower_bound A >= upper_bound
divset(D,i)-lower_bound divset(D,i) by A35,XREAL_1:10;
upper_bound divset(D,i)= D.i by A20,A34,INTEGRA1:def 4;
then upper_bound divset(D1,j)-lower_bound divset(D1,j) >=
upper_bound divset(D,i)-lower_bound A by A25,A24,XREAL_1:9;
then upper_bound divset(D1,j)-lower_bound divset(D1,j) >=
upper_bound divset(D,i)-lower_bound divset( D,i) by A36,XXREAL_0:2;
then vol(divset(D1,j)) >= upper_bound divset(D,i)-lower_bound
divset(D,i) by INTEGRA1:def 5;
then
A37: vol(divset(D1,j)) >= vol(divset(D,i)) by INTEGRA1:def 5;
i in Seg len D by A20,FINSEQ_1:def 3;
then i in Seg len upper_volume(chi(A,A),D) by INTEGRA1:def 6;
then
A38: i in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3;
vol(divset(D,i)) = upper_volume(chi(A,A),D).i by A20,
INTEGRA1:20;
then vol(divset(D,i)) in rng upper_volume(chi(A,A),D) by A38,
FUNCT_1:def 3;
then
A39: vol
(divset(D,i))>=min rng upper_volume(chi(A,A),D) by XXREAL_2:def 7;
vol(divset(D1,j)) <= delta(D1) by A21,Lm5;
then delta(D1) >= vol(divset(D,i)) by A37,XXREAL_0:2;
hence thesis by A39,XXREAL_0:2;
end;
end;
hence contradiction by A11;
end;
A40: P[1]
proof
len D in Seg len D by FINSEQ_1:3;
then 1 <= len D by FINSEQ_1:1;
then
A41: 1 in dom D by FINSEQ_3:25;
then consider j such that
A42: j in dom D1 and
A43: D.1 in divset(D1,j) by Th3,INTEGRA1:6;
PLf(D2,indx(D2,D1,j))-PLf(D1,j) <=1*(upper_bound rng f-
lower_bound rng f)*delta(D1)
proof
A44: j <> 1 by A19,A41,A42,A43;
then reconsider j1=j-1 as Element of NAT by A42,INTEGRA1:7;
A45: j1 in dom D1 by A42,A44,INTEGRA1:7;
then j1 in Seg len D1 by FINSEQ_1:def 3;
then j1 in Seg len lower_volume(f,D1) by INTEGRA1:def 7;
then
A46: j1 in dom lower_volume(f,D1) by FINSEQ_1:def 3;
A47: j-1 in dom D1 by A42,A44,INTEGRA1:7;
then
A48: indx(D2,D1,j1) in dom D2 by A13,INTEGRA1:def 19;
then
A49: indx(D2,D1,j1) in Seg len D2 by FINSEQ_1:def 3;
then
A50: 1 <= indx(D2,D1,j1) by FINSEQ_1:1;
then mid(D2,1,indx(D2,D1,j1)) is increasing by A48,INTEGRA1:35;
then
A51: D2|indx(D2,D1,j1) is increasing by A50,FINSEQ_6:116;
j < j+1 by NAT_1:13;
then j1 < j by XREAL_1:19;
then
A52: indx(D2,D1,j1) < indx(D2,D1,j) by A13,A42,A45,Th8;
then
A53: indx(D2,D1,j1)+1 <= indx(D2,D1,j) by NAT_1:13;
A54: Sum mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j
)) -Sum mid(lower_volume(f,D1),j,j) <= (upper_bound rng f-lower_bound rng f)*
delta(D1)
proof
A55: indx(D2,D1,j)-indx(D2,D1,j1) <= 2
proof
reconsider ID1=indx(D2,D1,j1)+1 as Element of NAT;
reconsider ID2=ID1+1 as Element of NAT;
assume indx(D2,D1,j)-indx(D2,D1,j1) > 2;
then
A56: indx(D2,D1,j1)+(1+1)<indx(D2,D1,j) by XREAL_1:20;
A57: ID1 < ID2 by NAT_1:13;
then indx(D2,D1,j1) <= ID2 by NAT_1:13;
then
A58: 1 <= ID2 by A50,XXREAL_0:2;
A59: indx(D2,D1,j) in dom D2 by A13,A42,INTEGRA1:def 19;
then
A60: indx(D2,D1,j) <= len D2 by FINSEQ_3:25;
then ID2 <= len D2 by A56,XXREAL_0:2;
then ID2 in Seg len D2 by A58,FINSEQ_1:1;
then
A61: ID2 in dom D2 by FINSEQ_1:def 3;
then
A62: D2.ID2<D2.indx(D2,D1, j) by A56,A59,SEQM_3:def 1;
A63: 1 <= ID1 by A50,NAT_1:13;
A64: D1.j = D2.indx(D2,D1,j) by A13,A42,INTEGRA1:def 19;
ID1 <= indx(D2,D1,j) by A56,A57,XXREAL_0:2;
then ID1 <= len D2 by A60,XXREAL_0:2;
then ID1 in Seg len D2 by A63,FINSEQ_1:1;
then
A65: ID1 in dom D2 by FINSEQ_1:def 3;
then
A66: D2.ID1<D2.ID2 by A57,A61,SEQM_3:def 1;
indx(D2,D1,j1) < ID1 by NAT_1:13;
then
A67: D2 .indx(D2,D1,j1)<D2.ID1 by A48,A65,SEQM_3:def 1;
A68: D1.j1 = D2.indx(D2,D1,j1) by A13,A45,INTEGRA1:def 19;
A69: not D2.ID1 in rng D1 & not D2.ID2 in rng D1
proof
assume
A70: D2.ID1 in rng D1 or D2.ID2 in rng D1;
per cases by A70;
suppose
D2.ID1 in rng D1;
then consider n such that
A71: n in dom D1 and
A72: D1.n=D2.ID1 by PARTFUN1:3;
j1<n by A45,A67,A68,A71,A72,SEQ_4:137;
then
A73: j<n+1 by XREAL_1:19;
D2.ID1<D2.indx(D2,D1,j) by A66,A62,XXREAL_0:2;
then n<j by A42,A64,A71,A72,SEQ_4:137;
hence contradiction by A73,NAT_1:13;
end;
suppose
D2.ID2 in rng D1;
then consider n such that
A74: n in dom D1 and
A75: D1.n=D2.ID2 by PARTFUN1:3;
D2.indx(D2,D1,j1)<D2.ID2 by A67,A66,XXREAL_0:2;
then j1<n by A45,A68,A74,A75,SEQ_4:137;
then
A76: j<n+1 by XREAL_1:19;
n<j by A42,A62,A64,A74,A75,SEQ_4:137;
hence contradiction by A76,NAT_1:13;
end;
end;
upper_bound divset(D1,j)=D1.j by A42,A44,INTEGRA1:def 4;
then
A77: upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A13,A42,
INTEGRA1:def 19;
lower_bound divset(D1,j)=D1.j1 by A42,A44,INTEGRA1:def 4;
then
A78: lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A13,A45,
INTEGRA1:def 19;
D2.ID2 in rng D \/ rng D1 by A14,A61,FUNCT_1:def 3;
then
A79: D2.ID2 in rng D by A69,XBOOLE_0:def 3;
D2.ID1 in rng D \/ rng D1 by A14,A65,FUNCT_1:def 3;
then
A80: D2.ID1 in rng D by A69,XBOOLE_0:def 3;
D2.indx(D2,D1,j1)<=D2.ID2 by A67,A66,XXREAL_0:2;
then D2.ID2 in divset(D1,j) by A62,A78,A77,INTEGRA2:1;
then
A81: D2.ID2 in rng D /\ divset(D1,j) by A79,XBOOLE_0:def 4;
D2.ID1<=D2.indx(D2,D1,j) by A66,A62,XXREAL_0:2;
then D2.ID1 in divset(D1,j) by A67,A78,A77,INTEGRA2:1;
then D2.ID1 in rng D /\ divset(D1,j) by A80,XBOOLE_0:def 4;
hence contradiction by A11,A42,A57,A65,A61,A81,Th5,SEQ_4:138;
end;
A82: 1 <= indx(D2,D1,j1)+1 by A50,NAT_1:13;
j <= len D1 by A42,FINSEQ_3:25;
then
A83: j <= len lower_volume(f,D1) by INTEGRA1:def 7;
A84: 1 <= j by A42,FINSEQ_3:25;
then
A85: mid(lower_volume(f,D1),j,j).1 = lower_volume(f,D1).j by A83,
FINSEQ_6:118;
reconsider lv = lower_volume(f,D1).j as Element of REAL
by XREAL_0:def 1;
j-'j+1 = 1 by Lm1;
then len mid(lower_volume(f,D1),j,j)= 1 by A84,A83,FINSEQ_6:118
;
then mid(lower_volume(f,D1),j,j) =<*lv*> by A85,FINSEQ_1:40;
then
A86: Sum mid(lower_volume(f,D1),j,j)= lower_volume(f,D1).j by
FINSOP_1:11;
indx(D2,D1,j) in dom D2 by A13,A42,INTEGRA1:def 19;
then
A87: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3;
then
A88: 1 <= indx(D2,D1,j) by FINSEQ_1:1;
indx(D2,D1,j) in Seg len lower_volume(f,D2) by A87,
INTEGRA1:def 7;
then
A89: indx(D2,D1,j) <= len lower_volume(f,D2) by FINSEQ_1:1;
then
A90: indx(D2,D1,j1)+1 <= len lower_volume(f, D2) by A53,XXREAL_0:2;
then indx(D2,D1,j1)+1 in Seg len lower_volume(f,D2) by A82,
FINSEQ_1:1;
then
A91: indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 7;
then
A92: indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1) =indx(D2,D1,j)-(indx(
D2,D1,j1)+1) by A53,XREAL_1:233;
then indx(D2,D1,j)-'(indx(D2,D1,j1) +1)+1 <= 2 by A55;
then
A93: len mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j))<=2 by A53,A88,A89,A82,A90,FINSEQ_6:118;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 0+1 by XREAL_1:6;
then
A94: 1 <= len mid(lower_volume(f,D2),(indx(D2, D1,j1)+1),indx
(D2,D1,j )) by A53,A88,A89,A82,A90,FINSEQ_6:118;
now
per cases by A94,A93,Lm2;
suppose
A95: len mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),
indx(D2,D1,j))=1;
upper_bound divset(D1,j)=D1.j by A42,A44,INTEGRA1:def 4;
then
A96: upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A13,A42,
INTEGRA1:def 19;
lower_bound divset(D1,j)=D1.j1 by A42,A44,INTEGRA1:def 4;
then lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A13,A45,
INTEGRA1:def 19;
then
A97: divset(D1,j)=[. D2.indx(D2,D1, j1),D2.indx(D2,D1,j)
.] by A96,INTEGRA1:4;
A98: delta(D1) >= 0 by Th9;
A99: upper_bound rng f - lower_bound rng f >= 0 by A1,Lm3,
XREAL_1:48;
A100: indx(D2,D1,j) in dom D2 by A13,A42,INTEGRA1:def 19;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=1 by A53,A88,A89,A82
,A90,A95,FINSEQ_6:118;
then
A101: indx(D2,D1,j)-(indx(D2,D1,j1)+1)=0 by A53,XREAL_1:233;
then indx(D2,D1,j)<>1 by A49,FINSEQ_1:1;
then
A102: upper_bound divset(D2,indx(D2,D1,j))=D2.indx(D2,D1,j
) by A100,INTEGRA1:def 4;
indx(D2,D1,j)-1=indx(D2,D1,j1) by A101;
then lower_bound divset(D2,indx(D2,D1,j)) =D2.indx(D2,D1,
j1) by A50,A101,A100,INTEGRA1:def 4;
then
A103: divset(D2,indx(D2,D1,j))=divset(D1,j) by A97,A102,
INTEGRA1:4;
reconsider lv = lower_volume(f,D2).(indx(D2,D1,j1)+1)
as Element of REAL
by XREAL_0:def 1;
mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)).1 =lower_volume(f,D2).(indx(D2,D1,j1)+1) by A88,A89,A82,A90,FINSEQ_6:118;
then mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)) =<*lv*> by A95,FINSEQ_1:40;
then Sum mid(lower_volume(f,D2),(indx( D2,D1,j1)+1),indx(
D2,D1,j )) =lower_volume(f,D2).(indx(D2,D1,j1)+1) by FINSOP_1:11
.=(lower_bound(rng(f|divset(D2,(indx(D2,D1,j1)+1)))))
*vol(divset(D2,(indx(D2,D1,j1)+1))) by A92,INTEGRA1:def 7
.=Sum mid(lower_volume(f,D1),j,j) by A42,A86,A101,A103,
INTEGRA1:def 7;
hence thesis by A98,A99;
end;
suppose
A104: len mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),
indx(D2,D1,j))=2;
A105: mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)).1 =lower_volume(f,D2).(indx(D2,D1,j1)+1) by A88,A89,A82,A90,FINSEQ_6:118;
A106: 2+(indx(D2,D1,j1)+1)>=0+1 by XREAL_1:7;
mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)).2 =LVf(D2).(2+(indx(D2,D1,j1)+1)-'1) by A53,A88,A89,A82,A90,A104,
FINSEQ_6:118
.=LVf(D2).(2+(indx(D2,D1,j1)+1)-1) by A106,XREAL_1:233
.=LVf(D2).(indx(D2,D1,j1)+(1+1));
then mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)) =<*lower_volume(f,D2).(indx(D2,D1,j1)+1), lower_volume(f,D2).(indx(D2,D1,
j1)+2)*> by A104,A105,FINSEQ_1:44;
then
A107: Sum mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(
D2,D1,j)) =lower_volume(f,D2).(indx(D2,D1,j1)+1) +lower_volume(f,D2).(indx(D2,
D1,j1)+2) by RVSUM_1:77;
A108: vol(divset(D2,indx(D2,D1,j1)+1))>=0 by INTEGRA1:9;
upper_bound divset(D1,j)=D1.j by A42,A44,INTEGRA1:def 4;
then
A109: upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A13,A42,
INTEGRA1:def 19;
A110: vol(divset(D2,indx(D2,D1,j1)+2))>=0 by INTEGRA1:9;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A53,A88,A89,A82
,A90,A104,FINSEQ_6:118;
then
A111: indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A53,XREAL_1:233;
then
A112: indx (D2,D1,j1)+2 in dom D2 by A13,A42,INTEGRA1:def 19;
lower_bound divset(D1,j)=D1.j1 by A42,A44,INTEGRA1:def 4;
then lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A13,A45,
INTEGRA1:def 19;
then
A113: vol(divset(D1,j)) =D2.(indx(D2,D1,j1)+2)-D2.(indx(D2
,D1,j1)+1) +D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A109,A111,INTEGRA1:def 5
;
indx(D2,D1,j1)+1 in Seg len lower_volume(f,D2) by A82,A90,
FINSEQ_1:1;
then indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 7;
then
A114: indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3;
A115: indx(D2,D1,j1)+1 <> 1 by A50,NAT_1:13;
then
A116: upper_bound divset(D2,(indx( D2,D1,j1)+1))= D2.(indx
(D2,D1,j1)+1) by A114,INTEGRA1:def 4;
indx(D2,D1,j1)+1-1=indx(D2,D1,j1)+0;
then
A117: lower_bound divset(D2,(indx(D2,D1,j1)+1))= D2.indx(
D2,D1,j1) by A114,A115,INTEGRA1:def 4;
A118: indx(D2,D1,j1)+1+1 > 1 by A82,NAT_1:13;
indx(D2,D1,j1)+2-1=indx(D2,D1,j1)+1;
then
A119: lower_bound divset(D2,(indx(D2,D1,j1)+2))= D2.(indx(
D2,D1,j1)+1) by A112,A118,INTEGRA1:def 4;
upper_bound divset(D2,(indx(D2,D1,j1)+2))= D2.(indx(
D2,D1,j1)+2) by A112,A118,INTEGRA1:def 4;
then vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+2))
+D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A119,A113,INTEGRA1:def 5
.=vol(divset(D2,indx(D2,D1,j1)+2)) +(upper_bound
divset(D2,indx(D2,D1,j1)+1)-lower_bound divset(D2,indx(D2,D1,j1)+1)) by A117
,A116;
then
A120: vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+1))+
vol(divset(D2,indx(D2,D1,j1)+2)) by INTEGRA1:def 5;
then
A121: lower_volume(f,D1).j=(lower_bound(rng(f| divset(D1,j
))))* (vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2))) by
A42,INTEGRA1:def 7;
A122: Sum mid(LVf(D2),indx(D2,D1,j1)+1,indx(D2,D1,j))-Sum
mid(LVf(D1) ,j,j) <=(upper_bound rng f - lower_bound rng f)*(vol(divset(D2,
indx(D2,D1,j1)+2)) +vol(divset(D2,indx(D2,D1,j1)+1)))
proof
set ID2=indx(D2,D1,j1)+2;
set ID1=indx(D2,D1,j1)+1;
set B = vol(divset(D2,ID1));
set C = vol(divset(D2,ID2));
divset(D1,j) c=A by A42,INTEGRA1:8;
then
A123: lower_bound rng(f|divset(D1,j)) >= lower_bound rng
f by A1,Lm4;
ID1 in dom D2 by A91,FINSEQ_1:def 3;
then divset(D2,ID1)c=A by INTEGRA1:8;
then lower_bound rng(f|divset(D2,ID1)) <= upper_bound
rng f by A1,Lm4;
then
A124: (lower_bound rng(f|divset(D2,ID1)))* vol(divset(D2
,ID1)) <=(upper_bound rng f)*vol(divset(D2,ID1)) by A108,XREAL_1:64;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A53,A88,A89,A82
,A90,A104,FINSEQ_6:118;
then
A125: indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A53,XREAL_1:233;
A126: indx(D2,D1,j) in dom D2 by A13,A42,INTEGRA1:def 19;
then divset(D2,ID2)c=A by A125,INTEGRA1:8;
then
A127: lower_bound rng(f|divset(D2,ID2)) <= upper_bound
rng f by A1,Lm4;
reconsider A = lower_bound(rng(f|divset(D1,j))) as Real;
A128: lower_volume(f,D1).j-A*B=A*C by A121;
(lower_bound rng(f|divset(D1,j)))*vol( divset(D2,
ID2)) >=(lower_bound rng f)*vol(divset(D2,ID2)) by A110,A123,XREAL_1:64;
then Sum mid(LVf(D1),j,j)>=(lower_bound rng(f|divset(D1
,j)))* vol(divset(D2, ID1)) +(lower_bound rng f)*vol(divset(D2,ID2)) by A86
,A128,XREAL_1:19;
then
A129: Sum mid(LVf(D1),j,j)-(lower_bound rng f)* vol(
divset(D2,ID2) ) >=(lower_bound rng(f|divset(D1,j)))*vol(divset(D2,ID1)) by
XREAL_1:19;
(lower_bound rng(f|divset(D1,j)))*vol(divset(D2,
ID1)) >=(lower_bound rng f)*vol(divset(D2,ID1)) by A108,A123,XREAL_1:64;
then Sum mid(LVf(D1),j,j)-(lower_bound rng f)* vol(
divset(D2,ID2) ) >=(lower_bound rng f)*vol(divset(D2,ID1)) by A129,XXREAL_0:2;
then
A130: Sum mid(LVf(D1),j,j) >=(lower_bound rng f)*vol(
divset(D2,ID2))+(lower_bound rng f)* vol(divset(D2,ID1)) by XREAL_1:19;
Sum mid(LVf(D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))
=(lower_bound rng(f|divset(D2,indx(D2,D1,j1)+2))) *vol(divset(D2,indx(D2,D1,j1)
+2)) +LVf(D2).(indx(D2,D1,j1)+1) by A107,A126,A125,INTEGRA1:def 7
.=(lower_bound rng(f|divset(D2,indx(D2,D1,j1)+2))) *
vol(divset(D2,indx(D2,D1,j1)+2)) +(lower_bound rng(f|divset(D2,indx(D2,D1,j1)+1
))) *vol(divset(D2,indx(D2,D1,j1)+1)) by A92,INTEGRA1:def 7;
then Sum mid(LVf(D2),ID1,indx(D2,D1,j)) -(lower_bound
rng(f|divset(D2,ID1)))*vol(divset(D2,ID1)) <=(upper_bound rng f)*vol(divset(D2,
ID2)) by A110,A127,XREAL_1:64;
then Sum mid(LVf(D2),ID1,indx(D2,D1,j)) <=(upper_bound
rng f)*vol(divset(D2,ID2)) +(lower_bound rng(f|divset(D2,ID1)))*vol(divset(D2,
ID1)) by XREAL_1:20;
then Sum mid(LVf(D2),ID1,indx(D2,D1,j))-(upper_bound
rng f)* vol(divset(D2, ID2)) <=(lower_bound rng(f|divset(D2,ID1)))*vol(divset(
D2,ID1)) by XREAL_1:20;
then Sum mid(LVf(D2),ID1,indx(D2,D1,j))-(upper_bound
rng f)* vol(divset(D2, ID2)) <=(upper_bound rng f)*vol(divset(D2,ID1)) by A124,
XXREAL_0:2;
then Sum mid(LVf(D2),ID1,indx(D2,D1,j))<=(upper_bound
rng f)*vol(divset(D2,ID2 ))+ (upper_bound rng f)*vol(divset(D2,ID1)) by
XREAL_1:20;
then Sum mid(LVf(D2),ID1,indx(D2,D1,j))-Sum mid(LVf(D1)
,j,j) <=(upper_bound rng f)*vol(divset(D2,ID2))+(upper_bound rng f)* vol(divset
(D2,ID1)) -((lower_bound rng f)*vol(divset(D2,ID2))+(lower_bound rng f)* vol(
divset(D2,ID1))) by A130,XREAL_1:13;
hence thesis;
end;
upper_bound rng f - lower_bound rng f >= 0 by A1,Lm3,
XREAL_1:48;
then (upper_bound rng f - lower_bound rng f)*( vol(divset
(D1,j))) <=(upper_bound rng f - lower_bound rng f)*delta(D1) by A42,Lm5,
XREAL_1:64;
hence thesis by A120,A122,XXREAL_0:2;
end;
end;
hence thesis;
end;
j < j+1 by NAT_1:13;
then
A131: j1 < j by XREAL_1:19;
indx(D2,D1,j) in dom D2 by A13,A42,INTEGRA1:def 19;
then
A132: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3;
then
A133: 1 <= indx(D2,D1,j) by FINSEQ_1:1;
A134: indx(D2,D1,j1) <= len D2 by A49,FINSEQ_1:1;
then
A135: len (D2|indx(D2,D1,j1))=indx(D2,D1,j1) by FINSEQ_1:59;
A136: j1 in Seg len D1 by A47,FINSEQ_1:def 3;
then
A137: j1 <= len D1 by FINSEQ_1:1;
for x1 being object st x1 in rng(D1|j1)
holds x1 in rng(D2|indx(D2,D1,j1))
proof
let x1 be object;
assume x1 in rng(D1|j1);
then consider k such that
A138: k in dom(D1|j1) and
A139: x1=(D1|j1).k by PARTFUN1:3;
k in Seg len(D1|j1) by A138,FINSEQ_1:def 3;
then
A140: k in Seg j1 by A137,FINSEQ_1:59;
then
A141: k in dom D1 by A45,RFINSEQ:6;
k <= j1 by A140,FINSEQ_1:1;
then D1.k <= D1.j1 by A47,A141,SEQ_4:137;
then D2.indx(D2,D1,k) <= D1.j1 by A13,A141,INTEGRA1:def 19;
then
A142: D2.indx(D2,D1,k)<=D2.indx(D2,D1,j1) by A13,A47,INTEGRA1:def 19;
A143: (D1|j1).k = D1.k by A45,A140,RFINSEQ:6;
D1.k in rng D1 by A141,FUNCT_1:def 3;
then x1 in rng D2 by A14,A139,A143,XBOOLE_0:def 3;
then consider n such that
A144: n in dom D2 and
A145: x1=D2.n by PARTFUN1:3;
D2.indx(D2,D1,k)=D2.n by A13,A139,A143,A141,A145,
INTEGRA1:def 19;
then
A146: n <= indx(D2,D1,j1) by A48,A144,A142,SEQM_3:def 1;
1 <= n by A144,FINSEQ_3:25;
then
A147: n in Seg indx(D2,D1,j1) by A146,FINSEQ_1:1;
then n in Seg len(D2|indx(D2,D1,j1)) by A134,FINSEQ_1:59;
then
A148: n in dom(D2|indx(D2,D1,j1)) by FINSEQ_1:def 3;
D2.n = (D2|indx(D2,D1,j1)).n by A48,A147,RFINSEQ:6;
hence thesis by A145,A148,FUNCT_1:def 3;
end;
then
A149: rng(D1|j1) c= rng(D2|indx(D2,D1,j1));
A150: 1 <= j1 by A136,FINSEQ_1:1;
lower_bound divset(D1,j) <= D.1 by A43,INTEGRA2:1;
then
A151: D1.j1 <= D.1 by A42,A44,INTEGRA1:def 4;
for x1 being object st x1 in rng(D2|indx(D2,D1,j1))
holds x1 in rng(D1|j1)
proof
let x1 be object;
assume x1 in rng(D2|indx(D2,D1,j1));
then consider k such that
A152: k in dom(D2|indx(D2,D1,j1)) and
A153: x1=(D2|indx(D2,D1,j1)).k by PARTFUN1:3;
k in Seg len(D2|indx(D2,D1,j1)) by A152,FINSEQ_1:def 3;
then
A154: k in Seg indx(D2,D1,j1) by A134,FINSEQ_1:59;
then
A155: k in dom D2 by A48,RFINSEQ:6;
A156: len(D1|j1) = j1 by A137,FINSEQ_1:59;
k <= indx(D2,D1,j1) by A154,FINSEQ_1:1;
then D2.k <= D2.indx(D2,D1,j1) by A48,A155,SEQ_4:137;
then
A157: D2.k <= D1.j1 by A13,A47,INTEGRA1:def 19;
A158: D2.k in rng D1 implies D2.k in rng(D1|j1)
proof
assume D2.k in rng D1;
then consider m such that
A159: m in dom D1 and
A160: D2.k = D1.m by PARTFUN1:3;
m in Seg len D1 by A159,FINSEQ_1:def 3;
then
A161: 1 <= m by FINSEQ_1:1;
A162: m <= j1 by A45,A157,A159,A160,SEQM_3:def 1;
then m in Seg j1 by A161,FINSEQ_1:1;
then
A163: D2.k = (D1|j1).m by A45,A160,RFINSEQ:6;
m in dom (D1|j1) by A156,A161,A162,FINSEQ_3:25;
hence thesis by A163,FUNCT_1:def 3;
end;
A164: D2.k in rng D implies D2.k = D1.j1
proof
assume D2.k in rng D;
then consider n such that
A165: n in dom D and
A166: D2.k=D.n by PARTFUN1:3;
1 <= n by A165,FINSEQ_3:25;
then D.1 <= D2.k by A41,A165,A166,SEQ_4:137;
then D1.j1 <= D2.k by A151,XXREAL_0:2;
hence thesis by A157,XXREAL_0:1;
end;
A167: D2.k in rng D implies D2.k in rng(D1|j1)
proof
j1 in Seg len(D1|j1) by A150,A156,FINSEQ_1:1;
then j1 in dom(D1|j1) by FINSEQ_1:def 3;
then
A168: (D1|j1).j1 in rng(D1|j1) by FUNCT_1:def 3;
assume
A169: D2.k in rng D;
j1 in Seg j1 by A150,FINSEQ_1:1;
hence thesis by A45,A164,A169,A168,RFINSEQ:6;
end;
D2.k in rng D2 by A155,FUNCT_1:def 3;
hence thesis by A14,A48,A153,A154,A167,A158,RFINSEQ:6
,XBOOLE_0:def 3;
end;
then rng(D2|indx(D2,D1,j1)) c= rng (D1|j1);
then
A170: rng (D2|indx(D2,D1,j1)) = rng (D1|j1) by A149,XBOOLE_0:def 10;
mid(D1,1,j1) is increasing by A42,A44,A150,INTEGRA1:7,35;
then
A171: D1|j1 is increasing by A150,FINSEQ_6:116;
then
A172: D2|indx(D2,D1,j1)=D1|j1 by A51,A170,Th6;
A173: for k st 1 <= k & k <= j1 holds k=indx(D2,D1,k)
proof
let k;
assume that
A174: 1 <= k and
A175: k <= j1;
assume
A176: k<>indx(D2,D1,k);
now
per cases by A176,XXREAL_0:1;
suppose
A177: k > indx(D2,D1,k);
k <= len D1 by A137,A175,XXREAL_0:2;
then
A178: k in dom D1 by A174,FINSEQ_3:25;
then indx(D2,D1,k) in dom D2 by A13,INTEGRA1:def 19;
then indx(D2,D1,k) in Seg len D2 by FINSEQ_1:def 3;
then
A179: 1<=indx(D2,D1,k) by FINSEQ_1:1;
A180: indx(D2,D1,k) < j1 by A175,A177,XXREAL_0:2;
then
A181: indx(D2,D1,k) in Seg j1 by A179,FINSEQ_1:1;
indx(D2,D1,k)<= indx(D2,D1,j1) by A13,A45,A175,A178,Th7;
then indx(D2,D1,k) in Seg indx(D2,D1,j1) by A179,FINSEQ_1:1
;
then
A182: (D2|indx(D2,D1,j1)).indx(D2,D1,k)= D2.indx(D2,D1,k)
by A48,RFINSEQ:6;
indx(D2,D1,k) <= len D1 by A137,A180,XXREAL_0:2;
then indx(D2,D1,k) in dom D1 by A179,FINSEQ_3:25;
then
A183: D1.k > D1.indx(D2,D1,k) by A177,A178,SEQM_3:def 1;
D1.k=D2. indx(D2,D1,k) by A13,A178,INTEGRA1:def 19;
hence contradiction by A45,A172,A182,A183,A181,RFINSEQ:6;
end;
suppose
A184: k < indx(D2,D1,k);
k <= len D1 by A137,A175,XXREAL_0:2;
then
A185: k in dom D1 by A174,FINSEQ_3:25;
then indx(D2,D1,k) <= indx(D2,D1,j1) by A13,A45,A175,Th7;
then
A186: k <= indx(D2,D1,j1) by A184,XXREAL_0:2;
then k <= len D2 by A134,XXREAL_0:2;
then
A187: k in dom D2 by A174,FINSEQ_3:25;
k in Seg j1 by A174,A175,FINSEQ_1:1;
then
A188: D1.k = (D1|j1).k by A47,RFINSEQ:6;
indx(D2,D1,k) in dom D2 by A13,A185,INTEGRA1:def 19;
then
A189: D2.k < D2.indx(D2,D1,k) by A184,A187,SEQM_3:def 1;
A190: k in Seg indx(D2,D1,j1) by A174,A186,FINSEQ_1:1;
D1.k=D2. indx(D2,D1,k) by A13,A185,INTEGRA1:def 19;
hence contradiction by A48,A172,A188,A189,A190,RFINSEQ:6;
end;
end;
hence contradiction;
end;
A191: for k be Nat st 1 <= k & k <= len(lower_volume(f,D1)|j1)
holds (lower_volume(f,D1)|j1).k = (lower_volume(f,D2)|indx(D2,D1,j1)).k
proof
indx(D2,D1,j1) in Seg len D2 by A48,FINSEQ_1:def 3;
then indx(D2,D1,j1) in Seg len lower_volume(f,D2) by
INTEGRA1:def 7;
then
A192: indx(D2,D1,j1) in dom lower_volume(f,D2) by FINSEQ_1:def 3;
let k be Nat;
assume that
A193: 1 <= k and
A194: k <= len(lower_volume(f,D1)|j1);
reconsider k as Element of NAT by ORDINAL1:def 12;
A195: len(lower_volume(f,D1)) = len D1 by INTEGRA1:def 7;
then
A196: k <= j1 by A137,A194,FINSEQ_1:59;
then k <= len D1 by A137,XXREAL_0:2;
then
A197: k in Seg len D1 by A193,FINSEQ_1:1;
then
A198: k in dom D1 by FINSEQ_1:def 3;
then
A199: indx(D2,D1,k) in dom D2 by A13,INTEGRA1:def 19;
A200: k in Seg j1 by A193,A196,FINSEQ_1:1;
then indx(D2,D1,k) in Seg j1 by A173,A193,A196;
then
A201: indx(D2,D1,k) in Seg indx(D2,D1,j1) by A150,A173;
then indx(D2,D1,k)<=indx(D2,D1,j1) by FINSEQ_1:1;
then
A202: indx(D2,D1,k)<=len D2 by A134,XXREAL_0:2;
A203: D1.k=D2.indx (D2,D1,k) by A13,A198,INTEGRA1:def 19;
A204: lower_bound divset(D1,k)= lower_bound divset(D2,indx(D2,
D1,k)) & upper_bound divset(D1,k)=upper_bound divset(D2,indx(D2,D1,k))
proof
per cases;
suppose
A205: k=1;
then
A206: upper_bound divset(D1,k)=D1.k by A198,INTEGRA1:def 4;
A207: lower_bound divset(D1,k)= lower_bound A by A198,A205,
INTEGRA1:def 4;
indx(D2,D1,k)=1 by A150,A173,A205;
hence thesis by A199,A203,A207,A206,INTEGRA1:def 4;
end;
suppose
A208: k<>1;
then reconsider k1=k-1 as Element of NAT by A198,INTEGRA1:7
;
k <= k+1 by NAT_1:11;
then k1 <= k by XREAL_1:20;
then
A209: k1 <= j1 by A196,XXREAL_0:2;
A210: k-1 in dom D1 by A198,A208,INTEGRA1:7;
then k1 in Seg len D1 by FINSEQ_1:def 3;
then 1 <= k1 by FINSEQ_1:1;
then k1=indx(D2,D1,k1) by A173,A209;
then
A211: D2.(indx(D2,D1,k)-1)= D2.indx(D2,D1,k1) by A173,A193,A196;
A212: indx(D2,D1,k)<>1 by A173,A193,A196,A208;
then
A213: lower_bound divset(D2,indx(D2,D1,k))= D2.(indx(D2,D1
,k)-1) by A199,INTEGRA1:def 4;
A214: upper_bound divset(D2,indx(D2,D1,k))=D2.indx(D2,D1,k
) by A199,A212,INTEGRA1:def 4;
A215: upper_bound divset(D1,k)=D1.k by A198,A208,INTEGRA1:def 4;
lower_bound divset(D1,k)=D1.(k-1) by A198,A208,
INTEGRA1:def 4;
hence thesis by A13,A198,A215,A210,A213,A214,A211,
INTEGRA1:def 19;
end;
end;
divset(D2,indx(D2,D1,k))= [. lower_bound divset(D2,indx(
D2,D1,k)), upper_bound divset(D2,indx(D2,D1,k)).] by INTEGRA1:4;
then
A216: divset(D1,k)=divset(D2,indx(D2,D1,k)) by A204,INTEGRA1:4;
A217: k in dom D1 by A197,FINSEQ_1:def 3;
j1 in Seg len(lower_volume(f,D1)) by A45,A195,FINSEQ_1:def 3;
then j1 in dom(lower_volume(f,D1)) by FINSEQ_1:def 3;
then
A218: (lower_volume(f,D1)|j1).k = lower_volume(f,D1).k by A200,
RFINSEQ:6
.=(lower_bound(rng(f|divset(D2,indx(D2,D1,k)))))*vol(
divset(D2,indx(D2,D1,k))) by A217,A216,INTEGRA1:def 7;
1<=indx(D2,D1,k) by A173,A193,A196;
then indx(D2,D1,k) in Seg len D2 by A202,FINSEQ_1:1;
then
A219: indx(D2,D1,k) in dom D2 by FINSEQ_1:def 3;
(lower_volume(f,D2)|indx(D2,D1,j1)).k =(lower_volume(f,
D2)|indx(D2,D1,j1)).indx(D2,D1,k) by A173,A193,A196
.=lower_volume(f,D2).indx(D2,D1,k) by A201,A192,RFINSEQ:6
.=(lower_bound(rng(f|divset(D2,indx(D2,D1,k)))))*vol(
divset(D2,indx(D2,D1,k))) by A219,INTEGRA1:def 7;
hence thesis by A218;
end;
indx(D2,D1,j1) in dom D2 by A13,A47,INTEGRA1:def 19;
then indx(D2,D1,j1) <= len D2 by FINSEQ_3:25;
then
A220: indx(D2,D1,j1) <= len lower_volume(f,D2) by INTEGRA1:def 7;
j1 <= len D1 by A47,FINSEQ_3:25;
then
A221: j1 <= len lower_volume(f,D1) by INTEGRA1:def 7;
len (D2|indx(D2,D1,j1))=len (D1|j1) by A51,A171,A170,Th6;
then indx(D2,D1,j1) =j1 by A137,A135,FINSEQ_1:59;
then len(lower_volume(f,D1)|j1)=indx(D2,D1,j1) by A221,
FINSEQ_1:59;
then len(lower_volume(f,D1)|j1)=len(lower_volume(f, D2)|indx(D2
,D1,j1)) by A220,FINSEQ_1:59;
then
A222: lower_volume(f,D2)|indx(D2,D1,j1)=lower_volume (f,D1)|j1
by A191,FINSEQ_1:14;
A223: j in Seg len D1 by A42,FINSEQ_1:def 3;
then
A224: 1 <= j by FINSEQ_1:1;
indx(D2,D1,j) in Seg len LVf(D2) by A132,INTEGRA1:def 7;
then
A225: indx(D2,D1,j) in dom LVf(D2) by FINSEQ_1:def 3;
indx(D2,D1,j) <= len D2 by A132,FINSEQ_1:1;
then
A226: indx(D2,D1,j) <= len LVf(D2) by INTEGRA1:def 7;
j in Seg len LVf(D1) by A223,INTEGRA1:def 7;
then
A227: j in dom LVf(D1) by FINSEQ_1:def 3;
j <= len D1 by A223,FINSEQ_1:1;
then
A228: j <= len LVf(D1) by INTEGRA1:def 7;
j1 in Seg len D1 by A45,FINSEQ_1:def 3;
then j1 in Seg len LVf(D1) by INTEGRA1:def 7;
then j1 in dom LVf(D1) by FINSEQ_1:def 3;
then PLf(D1,j1)=Sum(LVf(D1)|j1) by INTEGRA1:def 20;
then PLf(D1,j1)+Sum mid(LVf(D1),j,j) =Sum((LVf(D1)|j1)^mid(LVf(
D1),j,j)) by RVSUM_1:75
.=Sum(mid(LVf(D1),1,j1)^mid(LVf(D1),j1+1,j)) by A150,
FINSEQ_6:116
.=Sum(mid(LVf(D1),1,j)) by A150,A228,A131,INTEGRA2:4
.=Sum(LVf(D1)|j) by A224,FINSEQ_6:116;
then
A229: PLf(D1,j1)+Sum mid(lower_volume(f,D1),j,j)=PLf(D1,j) by A227,
INTEGRA1:def 20;
indx(D2,D1,j1) in Seg len D2 by A48,FINSEQ_1:def 3;
then indx(D2,D1,j1) in Seg len LVf(D2) by INTEGRA1:def 7;
then indx(D2,D1,j1) in dom LVf(D2) by FINSEQ_1:def 3;
then PLf(D2,indx(D2,D1,j1)) =Sum(LVf(D2)|indx(D2,D1,j1)) by
INTEGRA1:def 20;
then PLf(D2,indx(D2,D1,j1)) +Sum mid(lower_volume(f,D2),(indx(
D2,D1,j1)+1),indx(D2,D1,j)) =Sum(LVf(D2)|indx(D2,D1,j1) ^mid(LVf(D2),(indx(D2,
D1,j1)+1),indx(D2,D1,j))) by RVSUM_1:75
.=Sum(mid(LVf(D2),1,indx(D2,D1,j1)) ^mid(LVf(D2),indx(D2,D1,
j1)+1,indx(D2,D1,j))) by A50,FINSEQ_6:116
.=Sum(mid(LVf(D2),1,indx(D2,D1,j))) by A50,A52,A226,INTEGRA2:4
.=Sum(LVf(D2)|indx(D2,D1,j)) by A133,FINSEQ_6:116;
then
A230: PLf(D2,indx(D2,D1,j1)) +Sum mid(lower_volume(f,D2),(indx(
D2,D1,j1)+1), indx(D2,D1,j)) =PLf(D2,indx(D2,D1,j)) by A225,INTEGRA1:def 20;
indx(D2,D1,j1) in Seg len D2 by A48,FINSEQ_1:def 3;
then indx(D2,D1,j1) in Seg len lower_volume(f,D2) by
INTEGRA1:def 7;
then indx(D2,D1,j1) in dom lower_volume(f,D2) by FINSEQ_1:def 3;
then PLf(D2,indx(D2,D1,j1))=Sum(lower_volume(f,D2)|indx(D2,D1,
j1)) by INTEGRA1:def 20
.=PLf(D1,j1) by A222,A46,INTEGRA1:def 20;
hence thesis by A54,A230,A229;
end;
hence thesis by A42,A43;
end;
reconsider i as non zero Element of NAT by A18,FINSEQ_1:1;
A231: for i being non zero Nat st P[i] holds P[i+1]
proof
let i be non zero Nat;
A232: i>=1 by NAT_1:14;
assume
A233: P[i];
P[i+1]
proof
A234: i <= i+1 by NAT_1:11;
assume
A235: i+1 in dom D;
then consider j such that
A236: j in dom D1 and
A237: D.(i+1) in divset(D1,j) by Th3,INTEGRA1:6;
A238: D2.indx(D2,D1,j)=D1.j by A13,A236,INTEGRA1:def 19;
i+1 <= len D by A235,FINSEQ_3:25;
then i <= len D by A234,XXREAL_0:2;
then
A239: i in Seg len D by A232,FINSEQ_1:1;
then
A240: i in dom D by FINSEQ_1:def 3;
consider n1 being Element of NAT such that
A241: n1 in dom D1 and
A242: D.i in divset(D1,n1) and
A243: PLf(D2,indx(D2,D1,n1))-PLf(D1,n1)<=i*(upper_bound rng
f- lower_bound rng f)*delta(D1) by A233,A239,FINSEQ_1:def 3;
A244: 1 <= n1+1 by NAT_1:12;
A245: n1 < j
proof
assume
A246: n1 >= j;
now
per cases by A246,XXREAL_0:1;
suppose
A247: n1=j;
D.i in rng D by A240,FUNCT_1:def 3;
then
A248: D.i in rng D /\ divset(D1,j) by A242,A247,XBOOLE_0:def 4;
D.(i+1) in rng D by A235,FUNCT_1:def 3;
then
A249: D.(i+1) in rng D /\ divset(D1,j) by A237,XBOOLE_0:def 4;
i+1 > i by XREAL_1:29;
hence contradiction by A11,A235,A236,A240,A248,A249,Th5,
SEQ_4:138;
end;
suppose
n1>j;
then
A250: n1>=j+1 by NAT_1:13;
then
A251: n1-1 >= j by XREAL_1:19;
1 <= j by A236,FINSEQ_3:25;
then 1+1 <= j+1 by XREAL_1:6;
then
A252: n1 <> 1 by A250,XXREAL_0:2;
lower_bound divset(D1,n1) <= D.i by A242,INTEGRA2:1;
then
A253: D.i >= D1.(n1-1) by A241,A252,INTEGRA1:def 4;
n1-1 in dom D1 by A241,A252,INTEGRA1:7;
then D1.j <= D1.(n1-1) by A236,A251,SEQ_4:137;
then
A254: D.i >= D1.j by A253,XXREAL_0:2;
A255: i < i+1 by XREAL_1:29;
A256: upper_bound divset(D1,j)=D1.j
proof
per cases;
suppose
j=1;
hence thesis by A236,INTEGRA1:def 4;
end;
suppose
j<>1;
hence thesis by A236,INTEGRA1:def 4;
end;
end;
D.(i+1)<=upper_bound divset(D1,j) by A237,INTEGRA2:1;
then D.i >= D.(i+1) by A256,A254,XXREAL_0:2;
hence contradiction by A235,A240,A255,SEQM_3:def 1;
end;
end;
hence thesis;
end;
then
A257: n1+1 <= j by NAT_1:13;
A258: 1 <= n1 by A241,FINSEQ_3:25;
A259: indx(D2,D1,n1) in dom D2 by A13,A241,INTEGRA1:def 19;
then
A260: 1 <= indx(D2,D1,n1) by FINSEQ_3:25;
A261: indx(D2,D1,j) in dom D2 by A13,A236,INTEGRA1:def 19;
then
A262: 1 <= indx(D2,D1,j) by FINSEQ_3:25;
A263: indx(D2,D1,j) <= len D2 by A261,FINSEQ_3:25;
then
A264: indx(D2,D1,j) <= len LVf(D2) by INTEGRA1:def 7;
A265: 1 <= j by A236,FINSEQ_3:25;
A266: j <= len D1 by A236,FINSEQ_3:25;
then
A267: n1+1 <= len D1 by A257,XXREAL_0:2;
then
A268: n1+1 in dom D1 by A244,FINSEQ_3:25;
then
A269: indx(D2,D1,n1+1) in dom D2 by A13,INTEGRA1:def 19;
then
A270: 1 <= indx(D2,D1,n1+1) by FINSEQ_3:25;
A271: D2.indx(D2,D1,n1+ 1)=D1.(n1+1) by A13,A268,INTEGRA1:def 19;
then D2.indx(D2,D1,n1+1) <= D2.indx(D2,D1,j) by A236,A257,A268
,A238,SEQ_4:137;
then
A272: indx(D2,D1,n1+1) <= indx(D2,D1,j) by A269,A261,SEQM_3:def 1;
then 1+indx(D2,D1,n1+1) <= indx(D2,D1,j)+1 by XREAL_1:6;
then 1 <= indx(D2,D1,j)+1-indx(D2,D1,n1+1) by XREAL_1:19;
then
A273: mid(D2,indx(D2,D1,n1+1),indx(D2,D1,j)).1 =D2.(1-1+indx(D2,
D1,n1+1)) by A272,A270,A263,FINSEQ_6:122
.=D1.(n1+1) by A13,A268,INTEGRA1:def 19;
A274: D2.indx(D2,D1,n1) = D1.n1 by A13,A241,INTEGRA1:def 19;
A275: j <= len LVf(D1) by A266,INTEGRA1:def 7;
then j in Seg len LVf(D1) by A265,FINSEQ_1:1;
then
A276: j in dom LVf(D1) by FINSEQ_1:def 3;
A277: indx(D2,D1,n1+1) <= len D2 by A269,FINSEQ_3:25;
n1 in Seg len D1 by A241,FINSEQ_1:def 3;
then n1 in Seg len LVf(D1) by INTEGRA1:def 7;
then n1 in dom LVf(D1) by FINSEQ_1:def 3;
then PLf(D1,n1)=Sum(LVf(D1)|n1) by INTEGRA1:def 20
.=Sum mid(LVf(D1),1,n1) by A258,FINSEQ_6:116;
then PLf(D1,n1)+Sum mid(LVf(D1),n1+1,j) =Sum(mid(LVf(D1),1,n1)
^mid(LVf(D1),n1+1,j)) by RVSUM_1:75
.=Sum mid(LVf(D1),1,j) by A245,A258,A275,INTEGRA2:4
.=Sum(LVf(D1)|j) by A265,FINSEQ_6:116;
then
A278: PLf(D1,j)=PLf(D1,n1)+Sum mid(LVf(D1),n1+1,j) by A276,
INTEGRA1:def 20;
indx(D2,D1,j) in Seg len D2 by A261,FINSEQ_1:def 3;
then indx(D2,D1,j) in Seg len LVf(D2) by INTEGRA1:def 7;
then
A279: indx(D2,D1,j) in dom LVf(D2) by FINSEQ_1:def 3;
A280: n1 >= 1 by A241,FINSEQ_3:25;
A281: j-n1 >= 1 by A257,XREAL_1:19;
Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) -Sum mid(
LVf(D1),n1+1,j) <= (upper_bound rng f-lower_bound rng f)*delta(D1)
proof
now
per cases by A257,XXREAL_0:1;
suppose
A282: n1+1=j;
A283: indx(D2,D1,j)-indx(D2,D1,n1)<=2
proof
A284: upper_bound divset(D1,j) = D1.j by A236,A245,A280,
INTEGRA1:def 4;
A285: lower_bound divset(D1,j) = D1.(j-1) by A236,A245,A280,
INTEGRA1:def 4;
A286: 1 <= indx(D2,D1,n1)+1 by A260,NAT_1:13;
assume indx(D2,D1,j) - indx(D2,D1,n1) > 2;
then
A287: indx(D2,D1,n1)+2 < indx(D2,D1, j) by XREAL_1:20;
then
A288: indx(D2,D1,n1)+2 <= len D2 by A263,XXREAL_0:2;
A289: indx(D2,D1,n1)+1 < indx(D2,D1,n1)+2 by XREAL_1:6;
then
A290: indx(D2,D1,n1) < indx(D2,D1,n1)+2 by NAT_1:13;
then 1 <= indx(D2,D1,n1)+2 by A260,XXREAL_0:2;
then
A291: indx(D2,D1,n1)+2 in dom D2 by A288,FINSEQ_3:25;
then
A292: D2.indx(D2,D1,j) >= D2.(indx(D2,D1,n1)+2) by A261,A287,
SEQ_4:137;
A293: not D2.(indx(D2,D1,n1)+2) in rng D1
proof
assume D2.(indx(D2,D1,n1)+2) in rng D1;
then consider k1 being Element of NAT such that
A294: k1 in dom D1 and
A295: D2.(indx(D2,D1,n1)+2) = D1.k1 by PARTFUN1:3;
D2.(indx(D2,D1,n1)+2) < D2.indx(D2,D1,j) by A261,A287
,A291,SEQM_3:def 1;
then
A296: k1 < j by A236,A238,A294,A295,SEQ_4:137;
D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+2) by A259,A290
,A291,SEQM_3:def 1;
then n1 < k1 by A241,A274,A294,A295,SEQ_4:137;
hence contradiction by A282,A296,NAT_1:13;
end;
D2.(indx(D2,D1,n1)+2) in rng D2 by A291,FUNCT_1:def 3;
then
A297: D2.(indx(D2,D1,n1)+2) in rng D by A14,A293,XBOOLE_0:def 3
;
A298: lower_bound divset(D1,j) = D1.(j-1) by A236,A245,A280,
INTEGRA1:def 4;
A299: upper_bound divset(D1,j) = D1.j by A236,A245,A280,
INTEGRA1:def 4;
D2.(indx(D2,D1,n1)+2) >= D2.indx(D2,D1,n1) by A259,A290
,A291,SEQ_4:137;
then D2.(indx(D2,D1,n1)+2) in divset(D1,j) by A274,A238
,A282,A298,A284,A292,INTEGRA2:1;
then
A300: D2.(indx(D2,D1,n1)+2) in rng D /\ divset(D1,j) by A297,
XBOOLE_0:def 4;
A301: indx(D2,D1,n1)+1 < indx(D2,D1,j) by A287,A289,XXREAL_0:2;
then indx(D2,D1,n1)+1 <= len D2 by A263,XXREAL_0:2;
then
A302: indx(D2,D1,n1)+1 in dom D2 by A286,FINSEQ_3:25;
then
A303: D2.indx(D2,D1,j) >= D2.(indx(D2,D1,n1)+1) by A261,A301,
SEQ_4:137;
A304: indx(D2,D1,n1) < indx(D2,D1,n1)+1 by NAT_1:13;
A305: not D2.(indx(D2,D1,n1)+1) in rng D1
proof
assume D2.(indx(D2,D1,n1)+1) in rng D1;
then consider k1 being Element of NAT such that
A306: k1 in dom D1 and
A307: D2.(indx(D2,D1,n1)+1) = D1.k1 by PARTFUN1:3;
D2.(indx(D2,D1,n1)+1) < D2.indx(D2,D1,j) by A261,A301
,A302,SEQM_3:def 1;
then
A308: k1 < j by A236,A238,A306,A307,SEQ_4:137;
D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+1) by A259,A304
,A302,SEQM_3:def 1;
then n1 < k1 by A241,A274,A306,A307,SEQ_4:137;
hence contradiction by A282,A308,NAT_1:13;
end;
D2.(indx(D2,D1,n1)+1) in rng D2 by A302,FUNCT_1:def 3;
then
A309: D2.(indx(D2,D1,n1)+1) in rng D by A14,A305,XBOOLE_0:def 3
;
D2.(indx(D2,D1,n1)+1) >= D2.indx(D2,D1,n1) by A259,A304
,A302,SEQ_4:137;
then D2.(indx(D2,D1,n1)+1) in divset(D1,j) by A274,A238
,A282,A285,A299,A303,INTEGRA2:1;
then D2.(indx(D2,D1,n1)+1) in rng D /\ divset(D1,j) by
A309,XBOOLE_0:def 4;
then D2.(indx(D2,D1,n1)+1) = D2.(indx(D2,D1,n1)+2) by A11
,A236,A300,Th5;
hence contradiction by A289,A302,A291,SEQM_3:def 1;
end;
A310: indx(D2,D1,n1)+1<indx(D2,D1,j) implies indx(D2,D1,n1
)+2=indx(D2, D1, j )
proof
assume indx(D2,D1,n1)+1 < indx(D2,D1,j);
then
A311: indx(D2,D1,n1)+1+1 <= indx (D2,D1,j) by NAT_1:13;
indx(D2,D1,n1)+2 >= indx(D2,D1,j) by A283,XREAL_1:20;
hence thesis by A311,XXREAL_0:1;
end;
A312: 1<=indx(D2,D1,n1)+1 by NAT_1:12;
A313: indx(D2,D1,j) <= len LVf(D2) by A263,INTEGRA1:def 7;
D1.n1 < D1.j by A236,A241,A245,SEQM_3:def 1;
then
A314: indx(D2,D1,n1)<indx(D2,D1,j) by A259,A274,A261,A238,
SEQ_4:137;
then
A315: indx(D2,D1,n1)+1 <= indx(D2,D1,j) by NAT_1:13;
then indx(D2,D1,n1)+1 <= len D2 by A263,XXREAL_0:2;
then indx(D2,D1,n1)+1 <= len LVf(D2) by INTEGRA1:def 7;
then
A316: len mid(LVf(D2),indx(D2,D1,n1)+1, indx(D2,D1,j)) =
indx(D2,D1,j)-'(indx(D2,D1,n1)+1)+1 by A262,A315,A312,A313,FINSEQ_6:118
.=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A315,XREAL_1:233
.=indx(D2,D1,j)-indx(D2,D1,n1);
indx(D2,D1,n1)+1 <= indx(D2,D1,j) by A314,NAT_1:13;
then
A317: indx(D2,D1,n1)+1 = indx(D2,D1, j) or indx(D2,D1,n1)+
1 < indx(D2,D1,j) by XXREAL_0:1;
A318: Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) <=(
upper_bound rng f)*vol(divset(D1,n1+1))
proof
per cases by A317,A310;
suppose
A319: indx(D2,D1,j)-indx(D2,D1,n1)=1;
A320: indx(D2,D1,n1)+1 > 1 by A260,NAT_1:13;
then upper_bound divset(D2,indx(D2,D1,n1)+1)=D2.(indx
(D2,D1,n1)+1) by A261,A319,INTEGRA1:def 4;
then
A321: upper_bound divset(D2,indx(D2,D1,n1)+1)=D1.j by A13
,A236,A319,INTEGRA1:def 19;
lower_bound divset(D2,indx(D2,D1,n1)+1)=D2.(indx
(D2,D1,n1)+1-1) by A261,A319,A320,INTEGRA1:def 4;
then
A322: lower_bound divset(D2,indx(D2, D1,n1)+1)=D1.n1
by A13,A241,INTEGRA1:def 19;
lower_bound divset(D1,n1+1)=D1.(n1+1-1) by A245,A280
,A268,A282,INTEGRA1:def 4;
then
A323: divset(D2,indx(D2,D1,n1)+1)=divset(D1,n1+1) by A245
,A280,A268,A282,A322,A321,INTEGRA1:def 4;
A324: vol(divset(D2,indx(D2,D1,n1)+1))>=0 by INTEGRA1:9;
reconsider LV = LVf(D2).(indx(D2,D1,n1)+1) as Element of REAL
by XREAL_0:def 1;
1=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A319;
then mid(LVf(D2),indx(D2,D1,n1) +1,indx(D2,D1,j)).1 =
LVf(D2).(1+(indx(D2,D1,n1)+1)-1) by A312,A313,FINSEQ_6:122
.=LVf(D2).(indx(D2,D1,n1)+1);
then
A325: mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))
=<*LV*> by A316,A319,FINSEQ_1:40;
LVf(D2).(indx(D2,D1,n1)+1) =(lower_bound rng(f|
divset(D2,indx(D2,D1,n1)+1))) *vol(divset(D2,indx(D2,D1,n1)+1)) by A261,A319,
INTEGRA1:def 7;
then LVf(D2).(indx(D2,D1,n1)+1) <=(upper_bound rng f)
*vol(divset(D1,n1+1)) by A1,A261,A319,A323,A324,Th18,XREAL_1:64;
hence thesis by A325,FINSOP_1:11;
end;
suppose
A326: indx(D2,D1,j)-indx(D2,D1,n1)=2;
indx(D2,D1,n1)+2 >= 2+1 by A260,XREAL_1:6;
then
A327: indx(D2,D1,n1)+2 <> 1;
then
A328: upper_bound divset(D2,indx(D2,D1,n1)+2)= D2.indx
(D2,D1,j) by A261,A326,INTEGRA1:def 4;
indx(D2,D1,n1)+2-1=indx(D2,D1,n1)+1;
then lower_bound divset(D2,indx(D2,D1,n1)+2) = D2.(
indx(D2,D1,n1)+1) by A261,A326,A327,INTEGRA1:def 4;
then
A329: vol(divset(D2,indx(D2,D1,n1)+2)) =D1.j-D2.(indx(
D2,D1,n1)+1) by A238,A328,INTEGRA1:def 5;
A330: upper_bound divset(D1,n1+1)=D1.(n1+1) by A245,A280,A268
,A282,INTEGRA1:def 4;
lower_bound divset(D1,n1+1)=D1.(n1+1-1) by A245,A280
,A268,A282,INTEGRA1:def 4;
then
A331: vol(divset(D1,n1+1))=D1.(n1+1)-D1.n1 by A330,
INTEGRA1:def 5;
A332: vol(divset(D2,indx(D2,D1,n1)+2)) >= 0 by INTEGRA1:9;
A333: indx(D2,D1,j) <= len LVf(D2) by A263,INTEGRA1:def 7;
A334: vol(divset(D2,indx(D2,D1,n1)+1)) >= 0 by INTEGRA1:9;
A335: 1 <= indx(D2,D1,n1)+1 by NAT_1:12;
A336: indx(D2,D1,n1)+1 <= indx(D2,D1,n1)+2 by XREAL_1:6;
then indx(D2,D1,n1)+1 <= len D2 by A263,A326,XXREAL_0:2
;
then
A337: indx(D2,D1,n1)+1 in dom D2 by A335,FINSEQ_3:25;
then LVf(D2).(indx(D2,D1,n1 )+1) =(lower_bound rng(f|
divset(D2,indx(D2,D1,n1)+1))) *vol(divset(D2,indx(D2,D1,n1)+1)) by
INTEGRA1:def 7;
then
A338: LVf(D2).(indx(D2,D1,n1)+1) <= (upper_bound rng f
)*vol(divset(D2,indx(D2, D1,n1)+1)) by A1,A337,A334,Th18,XREAL_1:64;
indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1=1+1 by A326;
then
A339: mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).2 =
LVf(D2).(2+(indx(D2,D1,n1)+1)-1) by A335,A336,A333,FINSEQ_6:122
.=LVf(D2).(indx(D2,D1,n1)+0+2);
indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1>=1 by A326;
then mid(LVf(D2),indx(D2,D1,n1) +1,indx(D2,D1,j)).1 =
LVf(D2).(1+(indx(D2,D1,n1)+1)-1) by A326,A335,A336,A333,FINSEQ_6:122
.=LVf(D2).(indx(D2,D1,n1)+1);
then mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =<*
LVf(D2).(indx(D2,D1,n1)+1),LVf(D2).(indx(D2,D1,n1)+2)*> by A316,A326,A339,
FINSEQ_1:44;
then
A340: Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))
=LVf(D2).(indx(D2,D1,n1)+1)+LVf(D2).(indx(D2,D1,n1)+2) by RVSUM_1:77;
A341: indx(D2,D1,n1)+1 > 1 by A260,NAT_1:13;
then
A342: upper_bound divset(D2,indx(D2,D1,n1)+1)=D2.(indx
(D2,D1,n1)+1) by A337,INTEGRA1:def 4;
lower_bound divset(D2,indx(D2,D1,n1)+1) = D2.(
indx(D2,D1,n1)+1-1) by A337,A341,INTEGRA1:def 4;
then
A343: vol(divset(D2,indx(D2,D1,n1)+1)) =D2.(indx(D2,D1
,n1)+1)-D1.n1 by A274,A342,INTEGRA1:def 5;
LVf(D2).(indx(D2,D1,n1 )+2) =(lower_bound rng(f|
divset(D2,indx(D2,D1,n1)+2))) *vol(divset(D2,indx(D2,D1,n1)+2)) by A261,A326,
INTEGRA1:def 7;
then LVf(D2).(indx(D2,D1,n1)+2) <= (upper_bound rng f
)*vol(divset(D2,indx(D2, D1,n1)+2)) by A1,A261,A326,A332,Th18,XREAL_1:64;
then Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))
<= (upper_bound rng f)*vol(divset(D2,indx(D2,D1,n1)+1)) + (upper_bound rng f)*
vol(divset(D2,indx(D2,D1,n1)+2)) by A340,A338,XREAL_1:7;
hence thesis by A282,A343,A329,A331;
end;
end;
A344: n1+1 <= len LVf(D1) by A267,INTEGRA1:def 7;
then
A345: len mid(LVf(D1),n1+1,j)=j-'(n1+1)+1 by A244,A282,
FINSEQ_6:118
.=j-j+1 by A282,XREAL_1:233
.= 1;
reconsider lv =(lower_bound rng(f|divset(D1,
n1+1)))* vol(divset(D1,n1+1)) as Element of REAL
by XREAL_0:def 1;
(n1+1)+1 <= j+1 by A257,XREAL_1:6;
then 1 <= j+1-(n1+1) by XREAL_1:19;
then mid(LVf(D1),n1+1,j).1 =LVf(D1).(1-1+(n1+1)) by A244
,A282,A344,FINSEQ_6:122
.=(lower_bound rng(f|divset(D1,n1+1)))* vol(divset(D1,
n1+1)) by A268,INTEGRA1:def 7;
then mid(LVf(D1),n1+1,j) =<*lv
*> by A345,FINSEQ_1:40;
then
A346: Sum mid(LVf(D1),n1+1,j) =(lower_bound rng(f|divset(
D1,n1+1)))*vol(divset(D1,n1+1)) by FINSOP_1:11;
divset(D1,n1+1) c= A by A268,INTEGRA1:8;
then
A347: lower_bound rng(f|divset(D1,n1+1)) >= lower_bound
rng f by A1,Lm4;
n1+1 in Seg len D1 by A268,FINSEQ_1:def 3;
then n1+1 in Seg len upper_volume(chi(A,A),D1) by
INTEGRA1:def 6;
then
A348: n1+1 in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3;
vol(divset(D1,n1+1))= upper_volume(chi(A,A),D1).(n1+
1) by A268,INTEGRA1:20;
then vol(divset(D1,n1+1)) in rng upper_volume(chi(A,A),D1
) by A348,FUNCT_1:def 3;
then
A349: vol(divset(D1,n1+1))<=delta(D1) by XXREAL_2:def 8;
upper_bound rng f-lower_bound rng f >= 0 by A1,Lm3,
XREAL_1:48;
then
A350: (upper_bound rng f-lower_bound rng f)* vol(divset(D1
,n1+1)) <=(upper_bound rng f-lower_bound rng f)*delta(D1) by A349,XREAL_1:64;
vol(divset(D1,n1+1)) >= 0 by INTEGRA1:9;
then Sum mid(LVf(D1),n1+1,j) >=(lower_bound rng f)*vol(
divset(D1,n1+1)) by A346,A347,XREAL_1:64;
then Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))-Sum
mid(LVf(D1),n1+1 ,j) <=(upper_bound rng f)*vol(divset(D1,n1+1))-(lower_bound
rng f)* vol(divset(D1,n1+1)) by A318,XREAL_1:13;
hence thesis by A350,XXREAL_0:2;
end;
suppose
A351: n1+1 < j;
A352: n1 < n1+1 by NAT_1:13;
then
A353: D1.n1 < D1.(n1+1) by A241,A268,SEQM_3:def 1;
then consider
B being non empty closed_interval Subset of REAL, MD1,MD2
being Division of B such that
A354: D1.n1=lower_bound B and
upper_bound B=MD2.(len MD2) and
A355: upper_bound B=MD1.(len MD1) and
A356: MD1 <= MD2 and
A357: MD1=mid(D1,n1+1,j) and
A358: MD2=mid(D2,indx(D2,D1,n1+1),indx(D2,D1,j)) by A13,A236,A257
,A268,A273,Th15;
A359: delta(MD1) >= 0 by Th9;
A360: len MD1 = j-'(n1+1)+1 by A257,A265,A266,A244,A267,A357,
FINSEQ_6:118;
then
A361: len MD1+(n1+1)-1 = j-(n1+1)+1+(n1+1)-1 by A257,XREAL_1:233
.=j;
j-'(n1+1) = j-(n1+1) by A257,XREAL_1:233;
then
A362: j-'(n1+1)+1=j-n1;
then
A363: len MD1 = j-n1 by A257,A265,A266,A244,A267,A357,
FINSEQ_6:118;
A364: B c= A
proof
let x1 be object;
A365: rng D1 c= A by INTEGRA1:def 2;
D1.n1 in rng D1 by A241,FUNCT_1:def 3;
then
A366: lower_bound A <= D1.n1 by A365,INTEGRA2:1;
assume
A367: x1 in B;
then reconsider x1 as Real;
A368: x1 <= MD1.(len MD1) by A355,A367,INTEGRA2:1;
D1.j in rng D1 by A236,FUNCT_1:def 3;
then
A369: D1.j <= upper_bound A by A365,INTEGRA2:1;
D1 .n1 <= x1 by A354,A367,INTEGRA2:1;
then
A370: lower_bound A <= x1 by A366,XXREAL_0:2;
MD1.(len MD1)=D1.(j-n1-1+(n1+1)) by A257,A281,A266,A244
,A357,A362,A363,FINSEQ_6:122
.=D1.j;
then x1 <= upper_bound A by A368,A369,XXREAL_0:2;
hence thesis by A370,INTEGRA2:1;
end;
then reconsider g=f|B as Function of B,REAL by FUNCT_2:32;
A371: len lower_volume(g,MD1)=len MD1 by INTEGRA1:def 7
.=j-'(n1+1)+1 by A257,A265,A266,A244,A267,A357,
FINSEQ_6:118
.=j-(n1+1)+1 by A257,XREAL_1:233;
A372: len MD1 in dom MD1 by FINSEQ_5:6;
then
A373: 1 <= len MD1 by FINSEQ_3:25;
A374: lower_bound divset(MD1,len MD1)=lower_bound divset(
D1,j) & upper_bound divset(MD1,len MD1)=upper_bound divset(D1,j)
proof
per cases;
suppose
A375: len MD1=1;
then
A376: upper_bound divset(MD1,len MD1)=MD1.(len MD1) by A372,
INTEGRA1:def 4;
A377: upper_bound divset(D1,j)=D1.j by A236,A245,A280,
INTEGRA1:def 4;
lower_bound divset(D1,j)=D1.(j-1) by A236,A245,A280,
INTEGRA1:def 4;
hence
thesis by A265,A266,A354,A357,A361,A372,A375,A376,A377,
FINSEQ_6:118,INTEGRA1:def 4;
end;
suppose
A378: len MD1<>1;
then len MD1-1 in dom MD1 by A372,INTEGRA1:7;
then
A379: len MD1-1 >= 1 by FINSEQ_3:25;
len MD1 <= len MD1+1 by NAT_1:11;
then
A380: len MD1 -1 <= len MD1 by XREAL_1:20;
upper_bound divset(MD1,len MD1)=MD1.(len MD1) by A372
,A378,INTEGRA1:def 4;
then
A381: upper_bound divset(MD1,len MD1) =D1.j by A257,A266,A244
,A357,A360,A361,A373,FINSEQ_6:122;
A382: len MD1-1+(n1+1)-1=j-1 by A363;
lower_bound divset(MD1,len MD1)= MD1.(len MD1-1)
by A372,A378,INTEGRA1:def 4;
then lower_bound divset(MD1,len MD1)=D1.(j-1) by A257
,A266,A244,A357,A360,A382,A379,A380,FINSEQ_6:122;
hence thesis by A236,A245,A280,A381,INTEGRA1:def 4;
end;
end;
A383: len MD1 in dom MD1 by FINSEQ_5:6;
A384: upper_bound divset(MD1,len MD1)=MD1.(len MD1)
proof
per cases;
suppose
len MD1 = 1;
hence thesis by A383,INTEGRA1:def 4;
end;
suppose
len MD1 <> 1;
hence thesis by A383,INTEGRA1:def 4;
end;
end;
D1.n1 < D1.(n1+1) by A241,A268,A352,SEQM_3:def 1;
then indx(D2,D1,n1) < indx(D2,D1,n1+1) by A259,A274,A269
,A271,SEQ_4:137;
then
A385: indx(D2,D1,n1)+1 <= indx(D2,D1,n1+ 1) by NAT_1:13;
then
A386: indx(D2,D1,n1)+1 <= len D2 by A277,XXREAL_0:2;
vol(B)=upper_bound B-D1.n1 by A354,INTEGRA1:def 5;
then vol(B)=D1.j-D1.n1 by A236,A245,A280,A355,A374,A384,
INTEGRA1:def 4;
then
A387: vol(B)<>0 by A236,A241,A245,SEQM_3:def 1;
A388: 1 <= indx(D2,D1,n1)+1 by A260,NAT_1:13;
A389: indx(D2,D1,n1) < indx(D2,D1,n1)+1 by NAT_1:13;
A390: indx(D2,D1,n1+1)=indx(D2,D1,n1)+1
proof
assume indx(D2,D1,n1+1)<> indx(D2,D1,n1)+1;
then
A391: indx(D2,D1,n1+1)>indx(D2,D1,n1 )+1 by A385,XXREAL_0:1;
A392: indx(D2,D1,n1)+1 in dom D2 by A388,A386,FINSEQ_3:25;
then
A393: D2.(indx(D2,D1,n1)+1) in rng D2 by FUNCT_1:def 3;
now
per cases by A14,A393,XBOOLE_0:def 3;
suppose
D2.(indx(D2,D1,n1)+1) in rng D1;
then consider n2 being Element of NAT such that
A394: n2 in dom D1 and
A395: D2.(indx(D2,D1,n1)+1) = D1.n2 by PARTFUN1:3;
D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+1) by A259
,A389,A392,SEQM_3:def 1;
then n1 < n2 by A241,A274,A394,A395,SEQ_4:137;
then
A396: n1+1 <= n2 by NAT_1:13;
D1.n2 < D1.(n1+1) by A269,A271,A391,A392,A395,
SEQM_3:def 1;
hence contradiction by A268,A394,A396,SEQ_4:137;
end;
suppose
A397: D2.(indx(D2,D1,n1)+1) in rng D;
A398: D.i <= upper_bound divset(D1,n1) by A242,INTEGRA2:1;
A399: upper_bound divset(D1,n1)=D1.n1
proof
per cases;
suppose
n1=1;
hence thesis by A241,INTEGRA1:def 4;
end;
suppose
n1<>1;
hence thesis by A241,INTEGRA1:def 4;
end;
end;
consider n2 being Element of NAT such that
A400: n2 in dom D and
A401: D2.(indx(D2,D1,n1)+1) = D.n2 by A397,PARTFUN1:3;
D1.n1 < D.n2 by A259,A274,A389,A392,A401,SEQM_3:def 1
;
then D.i < D.n2 by A398,A399,XXREAL_0:2;
then i < n2 by A240,A400,SEQ_4:137;
then
A402: i+1 <= n2 by NAT_1:13;
n1+1 +1 <= j by A351,NAT_1:13;
then
A403: n1+1 <= j-1 by XREAL_1:19;
j -1 in dom D1 by A236,A245,A280,INTEGRA1:7;
then
A404: D1.(n1+1) <= D1.(j-1) by A268,A403,SEQ_4:137;
A405: lower_bound divset(D1,j) <= D.(i+1) by A237,
INTEGRA2:1;
lower_bound divset(D1,j) = D1.(j-1) by A236,A245,A280
,INTEGRA1:def 4;
then
A406: D1.(n1+1) <= D.(i+1) by A404,A405,XXREAL_0:2;
D .n2 < D1.(n1+1) by A269,A271,A391,A392,A401,
SEQM_3:def 1;
then D.n2 < D.(i+1) by A406,XXREAL_0:2;
hence contradiction by A235,A400,A402,SEQ_4:137;
end;
end;
hence contradiction;
end;
A407: j <= len LVf(D1) by A266,INTEGRA1:def 7;
A408: for k be Nat st 1<=k & k<=len lower_volume(g,MD1)
holds lower_volume(g,MD1).k=mid(LVf(D1),n1+1,j).k
proof
let k be Nat;
assume that
A409: 1 <= k and
A410: k <= len lower_volume(g,MD1);
k in Seg len lower_volume(g,MD1) by A409,A410,FINSEQ_1:1;
then
A411: k in Seg len MD1 by INTEGRA1:def 7;
then
A412: k in dom MD1 by FINSEQ_1:def 3;
k in dom MD1 by A411,FINSEQ_1:def 3;
then
A413: lower_volume(g,MD1).k =(lower_bound rng(g|divset(
MD1,k)))*vol(divset(MD1,k)) by INTEGRA1:def 7;
consider k2 being Element of NAT such that
A414: n1+1=1+k2;
A415: 1 <= k+k2 by A409,NAT_1:12;
k <= j-((n1+1)-1) by A371,A410;
then k+((n1+1)-1) <= j by XREAL_1:19;
then k+k2 <= len D1 by A266,A414,XXREAL_0:2;
then
A416: k+k2 in Seg len D1 by A415,FINSEQ_1:1;
then
A417: k+k2 in dom D1 by FINSEQ_1:def 3;
1+1 <= k+k2 by A258,A409,A414,XREAL_1:7;
then
A418: 1 < k+k2 by NAT_1:13;
A419: k2=(n1+1)-1 by A414;
A420: lower_bound divset(D1,k+k2)= lower_bound divset(
MD1,k) & upper_bound divset(D1,k+k2)= upper_bound divset(MD1,k)
proof
per cases;
suppose
A421: k=1;
then upper_bound divset(MD1,k) =MD1.k by A412,
INTEGRA1:def 4;
then
A422: upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A257
,A266,A244,A357,A371,A409,A410,FINSEQ_6:122;
lower_bound divset(MD1,k)=D1.n1 by A354,A412,A421,
INTEGRA1:def 4;
hence thesis by A419,A418,A417,A421,A422,
INTEGRA1:def 4;
end;
suppose
A423: k<>1;
then upper_bound divset(MD1,k)=MD1.k by A412,
INTEGRA1:def 4;
then
A424: upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A257
,A266,A244,A357,A371,A409,A410,FINSEQ_6:122;
A425: k -1 <= j-(n1+1)+1 by A371,A410,XREAL_1:146
,XXREAL_0:2;
A426: lower_bound divset(MD1,k)=MD1.(k-1) by A412,A423,
INTEGRA1:def 4;
A427: k-1 in dom MD1 by A412,A423,INTEGRA1:7;
then 1 <= k-1 by FINSEQ_3:25;
then lower_bound divset(MD1,k) =D1.(k-1+(n1+1)-1)
by A257,A266,A244,A357,A427,A425,A426,FINSEQ_6:122;
hence thesis by A414,A418,A417,A424,INTEGRA1:def 4;
end;
end;
divset(MD1,k)=[.lower_bound divset(MD1,k),
upper_bound divset(MD1,k).] by INTEGRA1:4;
then
A428: divset(D1,k+k2)=divset(MD1,k) by A420,INTEGRA1:4;
A429: k+k2 in dom D1 by A416,FINSEQ_1:def 3;
A430: mid(LVf(D1),n1+1,j).k =LVf(D1).(k+(n1+1)-1) by A257,A244
,A371,A407,A409,A410,FINSEQ_6:122
.=(lower_bound rng(f|divset(D1,k+k2)))* vol(divset(
D1,k+k2)) by A414,A429,INTEGRA1:def 7;
k in dom MD1 by A411,FINSEQ_1:def 3;
then divset(D1,k+k2) c= B by A428,INTEGRA1:8;
hence thesis by A413,A430,A428,FUNCT_1:51;
end;
A431: g|B is bounded
proof
consider a be Real such that
A432: for x being object st x in A /\ dom f holds a<=f.
x by A1,RFUNCT_1:71;
for x being object st x in B /\ dom g holds a<=g.x
proof
let x be object;
A433: dom f /\ B c= dom f /\ A by A364,XBOOLE_1:26;
assume x in B /\ dom g;
then
A434: x in dom g by XBOOLE_0:def 4;
then x in dom f /\ B by RELAT_1:61;
then a <= f.x by A432,A433;
hence thesis by A434,FUNCT_1:47;
end;
then
A435: g|B is bounded_below by RFUNCT_1:71;
consider a be Real such that
A436: for x being object st x in A /\ dom f holds f.x<=
a by A1,RFUNCT_1:70;
for x being object st x in B /\ dom g holds g.x<=a
proof
let x be object;
A437: dom f /\ B c= dom f /\ A by A364,XBOOLE_1:26;
assume x in B /\ dom g;
then
A438: x in dom g by XBOOLE_0:def 4;
then x in dom f /\ B by RELAT_1:61;
then a >= f.x by A436,A437;
hence thesis by A438,FUNCT_1:47;
end;
then g|B is bounded_above by RFUNCT_1:70;
hence thesis by A435;
end;
rng f is bounded_below by A1,INTEGRA1:11;
then
A439: lower_bound rng f <= lower_bound rng g by RELAT_1:70
,SEQ_4:47;
rng f is bounded_above by A1,INTEGRA1:13;
then upper_bound rng f >= upper_bound rng g by RELAT_1:70
,SEQ_4:48;
then upper_bound rng f-lower_bound rng f>= upper_bound
rng g - lower_bound rng g by A439,XREAL_1:13;
then
A440: (upper_bound rng f-lower_bound rng f)*delta(MD1)>= (
upper_bound rng g-lower_bound rng g)*delta(MD1 ) by A359,XREAL_1:64;
A441: n1 < j-1 by A351,XREAL_1:20;
A442: indx(D2,D1,j) <= len LVf(D2) by A263,INTEGRA1:def 7;
A443: len MD2=indx(D2,D1,j)-'indx(D2,D1,n1+1)+1 by A272,A270,A277
,A262,A263,A358,FINSEQ_6:118;
then
A444: len MD2 = indx(D2,D1,j)-indx(D2,D1,n1+1)+1 by A272,
XREAL_1:233;
then
A445: len lower_volume(g,MD2)=indx(D2,D1,j)-(indx(D2,D1,n1
)+1)+1 by A390,INTEGRA1:def 7;
for x1 being object holds
x1 in (rng MD1 \/ {D.(i+1)}) implies x1 in rng MD2
proof let x1 be object;
assume
A446: x1 in rng MD1 \/ {D.(i+1)};
then reconsider x1 as Real;
now
per cases by A446,XBOOLE_0:def 3;
suppose
A447: x1 in rng MD1;
rng MD1 <> {};
then 1 in dom MD1 by FINSEQ_3:32;
then
A448: 1 <= len MD1 by FINSEQ_3:25;
rng MD1 c= rng D1 by A357,FINSEQ_6:119;
then
A449: x1 in rng D1 by A447;
rng D1 c= rng D2 by A13,INTEGRA1:def 18;
then consider k such that
A450: k in dom D2 and
A451: D2.k = x1 by A449,PARTFUN1:3;
MD1.1=D1.(n1+1) by A265,A266,A244,A267,A357,
FINSEQ_6:118;
then D2.indx(D2,D1,n1+1 ) <= x1 by A271,A447,Th16;
then
A452: indx(D2,D1,n1+1)<=k by A269,A450,A451,SEQM_3:def 1;
then consider n being Nat such that
A453: k+1 = indx(D2,D1,n1+1)+n by NAT_1:10,12;
A454: len MD1=j-'(n1+1)+1 by A257,A265,A266,A244,A267,A357,
FINSEQ_6:118;
then len MD1+(n1+1)-1=j-(n1+1)+1+(n1+1)-1 by A257,
XREAL_1:233
.=j;
then MD1.(len MD1) =D1.j by A257,A266,A244,A357,A448
,A454,FINSEQ_6:122;
then x1 <= D2.indx(D2,D1,j) by A238,A447,Th16;
then k<=indx(D2,D1,j) by A261,A450,A451,SEQM_3:def 1;
then k-indx(D2,D1,n1+1)<=indx(D2,D1,j)-indx(D2,D1,
n1+1) by XREAL_1:9;
then
A455: k -indx(D2,D1,n1+1)+1<=indx(D2,D1,j)-indx(D2,
D1,n1+1)+1 by XREAL_1:6;
indx(D2,D1,n1+1)+1<=k+1 by A452,XREAL_1:6;
then
A456: 1 <= k+1-indx(D2, D1,n1+1) by XREAL_1:19;
then
A457: n in dom MD2 by A444,A455,A453,FINSEQ_3:25;
MD2.n=D2.(n+indx(D2,D1,n1+1)-1) by A272,A270
,A263,A358,A456,A455,A453,FINSEQ_6:122
.=D2.k by A453;
hence x1 in rng MD2 by A451,A457,FUNCT_1:def 3;
end;
suppose
x1 in {D.(i+1)};
then
A458: x1 = D.(i+1) by TARSKI:def 1;
reconsider j1 = j-1 as Element of NAT by A236,A245
,A280,INTEGRA1:7;
A459: rng D c= rng D2 by A12,INTEGRA1:def 18;
D.(i+1) in rng D by A235,FUNCT_1:def 3;
then consider k such that
A460: k in dom D2 and
A461: x1 = D2.k by A458,A459,PARTFUN1:3;
D.(i+1) <= upper_bound divset(D1,j) by A237,
INTEGRA2:1;
then x1 <= D1.j by A236,A245,A280,A458,INTEGRA1:def 4
;
then
A462: D2
.k <= D2.indx(D2,D1,j) by A13,A236,A461,INTEGRA1:def 19;
n1<j1 by A351,XREAL_1:20;
then
A463: n1+1 <= j1 by NAT_1:13;
j -1 in dom D1 by A236,A245,A280,INTEGRA1:7;
then
A464: D1.(n1+1) <= D1.(j-1) by A268,A463,SEQ_4:137;
lower_bound divset(D1,j) <= D.(i+1) by A237,
INTEGRA2:1;
then D1.(j-1) <= x1 by A236,A245,A280,A458,
INTEGRA1:def 4;
then D2.indx(D2,D1,n1+1) <= D2.k by A271,A461,A464,
XXREAL_0:2;
hence x1 in rng MD2 by A269,A261,A358,A460,A461,A462
,Th17;
end;
end;
hence thesis;
end;
then
A465: rng MD1 \/ {D.(i+1)} c= rng MD2;
rng MD2 <> {};
then 1 in dom MD2 by FINSEQ_3:32;
then
A466: 1 <= len MD2 by FINSEQ_3:25;
A467: len MD2-1+indx(D2,D1,n1+1)=indx(D2,D1,j) by A444;
for x1 being object holds
x1 in rng MD2 implies x1 in (rng MD1 \/ {D.(i+1)})
proof let x1 be object;
assume
A468: x1 in rng MD2;
then reconsider x1 as Real;
MD2.1=D2.indx(D2,D1,n1+1) by A270,A277,A262,A263,A358,
FINSEQ_6:118;
then
A469: D1 .(n1+1) <= x1 by A271,A468,Th16;
MD2.(len MD2)=D2.indx(D2,D1,j) by A272,A270,A263,A358
,A466,A443,A467,FINSEQ_6:122;
then
A470: x1 <= D1.j by A238,A468,Th16;
A471: rng MD2 c= rng D2 by A358,FINSEQ_6:119;
now
per cases by A14,A468,A471,XBOOLE_0:def 3;
suppose
x1 in rng D1;
then consider k such that
A472: k in dom D1 and
A473: D1.k = x1 by PARTFUN1:3;
A474: n1 +1 <= k by A268,A469,A472,A473,SEQM_3:def 1;
then
A475: 1 <= k-n1 by XREAL_1:19;
n1 <= n1+1 by NAT_1:11;
then consider n being Nat such that
A476: k=n1+n by A474,NAT_1:10,XXREAL_0:2;
A477: k <= j by A236,A470,A472,A473,SEQM_3:def 1;
then k-n1 <= len MD1 by A363,XREAL_1:9;
then n in dom MD1 by A475,A476,FINSEQ_3:25;
then
A478: MD1.n in rng MD1 by FUNCT_1:def 3;
j-(n1+1)+1=j-n1;
then
A479: k-n1 <= j-(n1+1)+1 by A477,XREAL_1:9;
MD1.n = D1.(k-n1-1+(n1+1)) by A257,A266,A244
,A357,A475,A479,A476,FINSEQ_6:122
.= D1.k;
hence x1 in (rng MD1 \/ {D.(i+1)}) by A473,A478,
XBOOLE_0:def 3;
end;
suppose
x1 in rng D;
then consider n such that
A480: n in dom D and
A481: D.n = x1 by PARTFUN1:3;
A482: not i+1 < n
proof
A483: upper_bound divset(D1,j)=D1.j
proof
per cases;
suppose
j=1;
hence thesis by A236,INTEGRA1:def 4;
end;
suppose
j<>1;
hence thesis by A236,INTEGRA1:def 4;
end;
end;
reconsider y1=D.(i+1) as Real;
A484: D.n in rng D by A480,FUNCT_1:def 3;
assume i+1 < n;
then
A485: D.(i+1) < D.n by A235,A480,SEQM_3:def 1;
lower_bound divset(D1,j) <= D.(i+1) by A237,
INTEGRA2:1;
then lower_bound divset(D1, j) <= D.n by A485,
XXREAL_0:2;
then D.n in divset(D1,j) by A470,A481,A483,
INTEGRA2:1;
then
A486: x1 in rng D /\ divset(D1,j) by A481,A484,
XBOOLE_0:def 4;
D.(i+1) in rng D by A235,FUNCT_1:def 3;
then y1 in rng D /\ divset(D1,j) by A237,
XBOOLE_0:def 4;
hence
contradiction by A11,A236,A481,A485,A486,Th5;
end;
A487: upper_bound divset(D1,n1)=D1.n1
proof
per cases;
suppose
n1=1;
hence thesis by A241,INTEGRA1:def 4;
end;
suppose
n1<>1;
hence thesis by A241,INTEGRA1:def 4;
end;
end;
D.i <= upper_bound divset(D1,n1) by A242,INTEGRA2:1;
then D.i < D1.(n1+1) by A353,A487,XXREAL_0:2;
then D.i < D.n by A469,A481,XXREAL_0:2;
then i < n by A240,A480,SEQ_4:137;
then i+1 <= n by NAT_1:13;
then i+1 = n or i+1 < n by XXREAL_0:1;
then x1 in {D.(i+1)} by A481,A482,TARSKI:def 1;
hence x1 in (rng MD1 \/ {D.(i+1)}) by XBOOLE_0:def 3;
end;
end;
hence thesis;
end;
then rng MD2 c= rng MD1 \/ {D.(i+1) };
then
A488: rng MD2 = rng MD1 \/ {D.(i+1)} by A465,XBOOLE_0:def 10;
delta(MD1) in rng upper_volume(chi(B,B),MD1) by XXREAL_2:def 8;
then consider k such that
A489: k in dom upper_volume(chi(B,B),MD1) and
A490: upper_volume(chi(B,B),MD1).k = delta(MD1) by PARTFUN1:3;
A491: k in Seg len upper_volume(chi(B,B),MD1) by A489,
FINSEQ_1:def 3;
then
A492: k in Seg len MD1 by INTEGRA1:def 6;
then
A493: k in dom MD1 by FINSEQ_1:def 3;
A494: k <= len MD1 by A492,FINSEQ_1:1;
then k+n1 <= j by A363,XREAL_1:19;
then
A495: k+n1 <= len D1 by A266,XXREAL_0:2;
A496: 1 <= k by A491,FINSEQ_1:1;
A497: n1+1>1 by A280,NAT_1:13;
then n1>1-1 by XREAL_1:19;
then
A498: k < k+n1 by XREAL_1:29;
then 1 < k+n1 by A496,XXREAL_0:2;
then
A499: k+n1 in dom D1 by A495,FINSEQ_3:25;
lower_bound divset(MD1,k)=lower_bound divset(D1,k+n1
) & upper_bound divset(MD1,k)=upper_bound divset(D1,k+n1)
proof
per cases;
suppose
A500: k=1;
then upper_bound divset(MD1,k)=MD1.k by A493,
INTEGRA1:def 4;
then
A501: upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A257,A266
,A244,A357,A360,A496,A494,FINSEQ_6:122;
lower_bound divset(D1,k+n1)=D1.(k+n1-1) by A496,A498
,A499,INTEGRA1:def 4;
hence thesis by A354,A497,A493,A499,A500,A501,
INTEGRA1:def 4;
end;
suppose
A502: k<>1;
then upper_bound divset(MD1,k)=MD1. k by A493,
INTEGRA1:def 4;
then
A503: upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A257,A266
,A244,A357,A360,A496,A494,FINSEQ_6:122;
A504: lower_bound divset(MD1,k)=MD1.(k-1) by A493,A502,
INTEGRA1:def 4;
A505: k-1 in dom MD1 by A493,A502,INTEGRA1:7;
then
A506: k-1 <= len MD1 by FINSEQ_3:25;
1 <= k-1 by A505,FINSEQ_3:25;
then lower_bound divset(MD1,k)=D1.(k-1+(n1+1)-1) by
A257,A266,A244,A357,A360,A505,A506,A504,FINSEQ_6:122;
hence thesis by A496,A498,A499,A503,INTEGRA1:def 4;
end;
end;
then divset(MD1,k)=[.lower_bound divset(D1,k+n1),
upper_bound divset(D1,k+n1).] by INTEGRA1:4;
then
A507: divset(MD1,k)=divset(D1,k+n1) by INTEGRA1:4;
k+n1 in Seg len D1 by A499,FINSEQ_1:def 3;
then k+n1 in Seg len upper_volume(chi(A,A),D1) by
INTEGRA1:def 6;
then
A508: k+n1 in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3;
k in dom MD1 by A492,FINSEQ_1:def 3;
then delta(MD1) = vol(divset(MD1,k)) by A490,INTEGRA1:20;
then
delta(MD1)=upper_volume(chi(A,A), D1).(k+n1) by A499,A507,INTEGRA1:20;
then delta(MD1) in rng upper_volume(chi(A,A),D1) by A508,
FUNCT_1:def 3;
then delta(MD1) <= max rng upper_volume(chi(A,A),D1) by
XXREAL_2:def 8;
then
A509: delta(MD1) <= delta(D1);
upper_bound rng f - lower_bound rng f >= 0 by A1,Lm3,
XREAL_1:48;
then
A510: (upper_bound rng f-lower_bound rng f)*delta(MD1) <=
(upper_bound rng f-lower_bound rng f)* delta(D1) by A509,XREAL_1:64;
lower_bound divset(D1,j) <= D.(i+1) by A237,INTEGRA2:1;
then
A511: D1.(j-1) <= D.(i+1) by A236,A245,A280,INTEGRA1:def 4;
A512: D.(i+1)<=upper_bound divset(D1,j) by A237,INTEGRA2:1;
A513: indx (D2,D1,n1)+1 <= indx(D2,D1,j) by A272,A385,XXREAL_0:2;
A514: for k be Nat st 1<=k & k<=len lower_volume(g,MD2)
holds lower_volume(g,MD2).k=mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k
proof
let k be Nat;
assume that
A515: 1 <= k and
A516: k <= len lower_volume(g,MD2);
A517: k in Seg len lower_volume(g,MD2) by A515,A516,FINSEQ_1:1;
A518: mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k =LVf
(D2).(k+(indx(D2,D1,n1)+1)-1) by A388,A445,A442,A513,A515,A516,FINSEQ_6:122;
1<=indx(D2,D1,n1)+1 by NAT_1:12;
then 1+1<=k+(indx(D2,D1,n1)+1) by A515,XREAL_1:7;
then
A519: 1 <= k+(indx(D2,D1,n1) +1)-1 by XREAL_1:19;
consider k2 being Element of NAT such that
A520: indx(D2,D1,n1)+1=1+k2;
k <= indx(D2,D1,j)-((indx(D2,D1,n1)+1)-1) by A444,A390
,A516,INTEGRA1:def 7;
then k+((indx(D2,D1,n1)+1)-1) <= indx(D2,D1,j) by
XREAL_1:19;
then k +(indx(D2,D1,n1)+1)-1 <= len LVf(D2) by A442,
XXREAL_0:2;
then k+k2 in Seg len LVf(D2) by A519,A520,FINSEQ_1:1;
then
A521: k+k2 in Seg len D2 by INTEGRA1:def 7;
then k+k2 in dom D2 by FINSEQ_1:def 3;
then
A522: mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k =(
lower_bound rng(f|divset(D2,k+k2)))* vol(divset(D2,k+k2)) by A518,A520,
INTEGRA1:def 7;
A523: k in Seg len MD2 by A517,INTEGRA1:def 7;
A524: lower_bound divset(MD2,k)=lower_bound divset(D2,k+
k2) & upper_bound divset(MD2,k)= upper_bound divset(D2,k+k2)
proof
k+k2>=1+1 by A260,A515,A520,XREAL_1:7;
then
A525: k+k2>1 by NAT_1:13;
A526: k in dom MD2 by A523,FINSEQ_1:def 3;
A527: k+k2 in dom D2 by A521,FINSEQ_1:def 3;
per cases;
suppose
A528: k=1;
then upper_bound divset(MD2,k)= MD2.k by A526,
INTEGRA1:def 4;
then
A529: upper_bound divset(MD2,k)=D2.(k+( indx(D2,D1,
n1)+1)-1) by A272,A263,A358,A388,A390,A445,A515,A516,FINSEQ_6:122;
A530: lower_bound divset(D2,k+k2)=D2.(k+k2-1) by A525,A527,
INTEGRA1:def 4;
lower_bound divset(MD2,k)=D1.n1 by A354,A526,A528,
INTEGRA1:def 4;
hence thesis by A13,A241,A520,A525,A527,A528,A529
,A530,INTEGRA1:def 4,def 19;
end;
suppose
A531: k<>1;
then upper_bound divset(MD2,k)=MD2.k by A526,
INTEGRA1:def 4;
then
A532: upper_bound divset(MD2,k)=D2.(k+( indx(D2,D1,
n1)+1)-1) by A272,A263,A358,A388,A390,A445,A515,A516,FINSEQ_6:122;
A533: k -1 <= indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A445
,A516,XREAL_1:146,XXREAL_0:2;
A534: lower_bound divset(MD2,k)=MD2.(k-1) by A526,A531,
INTEGRA1:def 4;
A535: k-1 in dom MD2 by A526,A531,INTEGRA1:7;
then 1 <= k-1 by FINSEQ_3:25;
then lower_bound divset(MD2,k)=D2.(k-1+(indx(D2,D1,
n1)+1)-1) by A272,A263,A358,A388,A390,A535,A533,A534,FINSEQ_6:122;
hence thesis by A520,A525,A527,A532,INTEGRA1:def 4;
end;
end;
divset (MD2,k)=[.lower_bound divset(MD2,k),
upper_bound divset(MD2,k).] by INTEGRA1:4;
then
A536: divset(MD2,k)=divset( D2,k+k2) by A524,INTEGRA1:4;
k in dom MD2 by A523,FINSEQ_1:def 3;
then divset(D2,k+k2) c= B by A536,INTEGRA1:8;
then
A537: rng(f|divset(D2,k+k2))=rng(g|divset(D2,k+k2)) by
FUNCT_1:51;
k in dom MD2 by A523,FINSEQ_1:def 3;
hence thesis by A522,A536,A537,INTEGRA1:def 7;
end;
lower_bound divset(D1,j)<=D.(i+1) by A237,INTEGRA2:1;
then
A538: D.(i+1) in divset(MD1,len MD1) by A374,A512,INTEGRA2:1;
j-1 in dom D1 by A236,A245,A280,INTEGRA1:7;
then D1.n1 <D1.(j-1) by A241,A441,SEQM_3:def 1;
then D.(i+1) > lower_bound B by A354,A511,XXREAL_0:2;
then Sum lower_volume(g,MD2)-Sum lower_volume(g,MD1)<= (
upper_bound rng g- lower_bound rng g)*delta(MD1) by A356,A431,A488,A538,A387
,Th13;
then
A539: Sum lower_volume(g,MD2)-Sum lower_volume(g,MD1)<= (
upper_bound rng f-lower_bound rng f)*delta(MD1) by A440,XXREAL_0:2;
indx(D2,D1,n1)+1 <= len LVf(D2) by A386,INTEGRA1:def 7;
then len mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =
indx(D2,D1,j)-'(indx(D2,D1,n1)+1)+1 by A262,A388,A442,A513,FINSEQ_6:118;
then len lower_volume(g,MD2) =len mid(LVf(D2),indx(D2,D1,
n1)+1,indx(D2,D1,j)) by A272,A385,A445,XREAL_1:233,XXREAL_0:2;
then
A540: Sum lower_volume(g,MD2)=Sum mid(LVf(D2),indx(D2,D1,
n1)+1,indx(D2,D1,j)) by A514,FINSEQ_1:14;
n1+1 <= len LVf(D1) by A267,INTEGRA1:def 7;
then len mid(LVf(D1),n1+1,j) =j-'(n1+1)+1 by A257,A265,A244
,A407,FINSEQ_6:118
.=j-(n1+1)+1 by A257,XREAL_1:233;
then Sum lower_volume(g,MD1)=Sum mid(LVf(D1),n1+1,j) by
A371,A408,FINSEQ_1:14;
hence thesis by A539,A510,A540,XXREAL_0:2;
end;
end;
hence thesis;
end;
then
A541: (PLf(D2,indx(D2,D1,n1))-PLf(D1,n1))+ (Sum mid(LVf(D2),
indx(D2,D1,n1)+1,indx(D2,D1,j)) -Sum mid(LVf(D1),n1+1,j)) <=i*(upper_bound rng
f-lower_bound rng f)*delta(D1)+ (upper_bound rng f-lower_bound rng f)*delta(D1)
by A243,XREAL_1:7;
n1 < n1+1 by NAT_1:13;
then D1.n1 < D1.(n1+1) by A241,A268,SEQM_3:def 1;
then indx(D2,D1,n1) < indx(D2,D1,n1+1) by A259,A274,A269,A271,
SEQ_4:137;
then
A542: indx(D2,D1,n1)<indx(D2,D1,j) by A272,XXREAL_0:2;
indx(D2,D1,n1) in Seg len D2 by A259,FINSEQ_1:def 3;
then indx(D2,D1,n1) in Seg len LVf(D2) by INTEGRA1:def 7;
then indx(D2,D1,n1) in dom LVf(D2) by FINSEQ_1:def 3;
then PLf(D2,indx(D2,D1,n1))=Sum(LVf(D2)|indx(D2,D1,n1)) by
INTEGRA1:def 20
.=Sum mid(LVf(D2),1,indx(D2,D1,n1)) by A260,FINSEQ_6:116;
then PLf(D2,indx(D2,D1,n1))+Sum mid(LVf(D2),indx(D2,D1,n1)+1,
indx(D2, D1,j)) =Sum(mid(LVf(D2),1,indx(D2,D1,n1))^ mid(LVf(D2),indx(D2,D1,n1)+
1,indx(D2,D1,j))) by RVSUM_1:75
.=Sum mid(LVf(D2),1,indx(D2,D1,j)) by A260,A542,A264,INTEGRA2:4
.=Sum(LVf(D2)|indx(D2,D1,j)) by A262,FINSEQ_6:116;
then PLf(D2,indx(D2,D1,j))= PLf(D2,indx(D2,D1,n1))+Sum mid(LVf
(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) by A279,INTEGRA1:def 20;
then (PLf(D2,indx(D2,D1,n1))-PLf(D1,n1))+ (Sum mid(LVf(D2),
indx(D2,D1,n1)+1,indx(D2,D1,j)) -Sum mid(LVf(D1),n1+1,j)) =PLf(D2,indx(D2,D1,j)
)-PLf(D1,j) by A278;
hence thesis by A236,A237,A541;
end;
hence thesis;
end;
for k being non zero Nat holds P[k] from NAT_1:sch 10(A40,
A231);
then P[i];
hence thesis by A17;
end;
A543: len D1 in dom D1 by FINSEQ_5:6;
then D1.(len D1) = D2.indx(D2,D1,len D1) by A13,INTEGRA1:def 19;
then upper_bound A = D2.indx(D2,D1,len D1) by INTEGRA1:def 2;
then
A544: D2.(len D2) = D2.indx(D2,D1,len D1) by INTEGRA1:def 2;
len D in dom D by FINSEQ_5:6;
then consider j such that
A545: j in dom D1 and
A546: D.(len D) in divset(D1,j) and
A547: PLf(D2,indx(D2,D1,j))-PLf(D1,j) <= (len D)*(upper_bound(rng
f)- lower_bound(rng f))*delta(D1) by A16;
A548: j = len D1
proof
assume
A549: j<>len D1;
j <= len D1 by A545,FINSEQ_3:25;
then j < len D1 by A549,XXREAL_0:1;
then D1.j < D1.(len D1) by A545,A543,SEQM_3:def 1;
then
A550: D1.j < upper_bound A by INTEGRA1:def 2;
A551: upper_bound divset(D1,j) < upper_bound A
proof
per cases;
suppose
j=1;
hence thesis by A545,A550,INTEGRA1:def 4;
end;
suppose
j<>1;
hence thesis by A545,A550,INTEGRA1:def 4;
end;
end;
D.(len D) <= upper_bound divset(D1,j) by A546,INTEGRA2:1;
hence contradiction by A551,INTEGRA1:def 2;
end;
indx(D2,D1,len D1) in dom D2 by A13,A543,INTEGRA1:def 19;
then indx(D2,D1,len D1)=len D2 by A15,A544,SEQ_4:138;
then PLf(D2,len D2)-lower_sum(f,D1)<=(len D)* (upper_bound(rng f)-
lower_bound(rng f))* delta(D1) by A547,A548,INTEGRA1:43;
hence thesis by INTEGRA1:43;
end;
hence thesis by A12,A13,A14;
end;
hence thesis;
end;
hence thesis;
end;
theorem Th22:
f|A is bounded implies
for D,D1 ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & 0<=
upper_sum(f,D)-upper_sum(f,D2) & 0<=upper_sum(f,D1)-upper_sum(f,D2)
proof
assume
A1: f|A is bounded;
let D,D1;
consider D2 such that
A3: D<=D2 and
A4: D1<=D2 and
A5: rng D2=rng D1 \/ rng D by Th4;
A6: upper_sum(f,D1)-upper_sum(f,D2)>=0 by A1,A4,INTEGRA1:45,XREAL_1:48;
upper_sum(f,D)-upper_sum(f,D2)>=0 by A1,A3,INTEGRA1:45,XREAL_1:48;
hence thesis by A3,A4,A5,A6;
end;
theorem Th23:
f|A is bounded implies
for D,D1 st delta(D1)<min rng upper_volume(chi(A,A),D) holds
ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D &
upper_sum(f,D1)-upper_sum(f,D2) <=
(len D)*(upper_bound(rng f)-lower_bound(rng f))*delta(D1)
proof
assume
A1: f|A is bounded; then
A2: for D,D1 ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & 0<=
upper_sum(f,D)-upper_sum(f,D2) & 0<=upper_sum(f,D1)-upper_sum(f,D2)
by Th22;
let D,D1;
assume
A8: delta(D1)<min rng upper_volume(chi(A,A),D);
ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & upper_sum(f,D1)-
upper_sum(f,D2)<=(len D)*(upper_bound(rng f)- lower_bound(rng f))*delta(D1)
proof
consider D2 be Division of A such that
A9: D<=D2 and
A10: D1<=D2 and
A11: rng D2=rng D1 \/ rng D and
0<=upper_sum(f,D)-upper_sum(f,D2) and
0<=upper_sum(f,D1)-upper_sum(f,D2) by A2;
upper_sum(f,D1)-upper_sum(f,D2) <= (len D)*(upper_bound(rng f)-
lower_bound(rng f))*delta(D1)
proof
deffunc UVf(Division of A) = upper_volume(f,$1);
deffunc PUf(Division of A,Nat) = (PartSums(upper_volume(f,$1))).$2;
A12: len D2 in dom D2 by FINSEQ_5:6;
A13: for i st i in dom D holds ex j st j in dom D1 & D.i in divset(D1,
j) & PUf(D1,j)-PUf(D2,indx(D2,D1,j))<=i*(upper_bound(rng f)- lower_bound(rng f)
)*delta(D1)
proof
defpred P[non zero Nat] means $1 in dom D implies ex j st j in dom
D1 & D.$1 in divset(D1,j) & PUf(D1,j)-PUf(D2,indx(D2,D1,j))<=$1*(upper_bound
rng f-lower_bound rng f) *delta(D1);
let i;
assume
A14: i in dom D;
then
A15: i in Seg len D by FINSEQ_1:def 3;
A16: for i,j st i in dom D & j in dom D1 & D.i in divset(D1,j) holds
j >= 2
proof
let i,j;
assume
A17: i in dom D;
assume that
A18: j in dom D1 and
A19: D.i in divset(D1,j);
assume j<2;
then j<1+1;
then
A20: j <= 1 by NAT_1:13;
j in Seg len D1 by A18,FINSEQ_1:def 3;
then j >= 1 by FINSEQ_1:1;
then j = 1 by A20,XXREAL_0:1;
then
A21: lower_bound divset(D1,j)=lower_bound A by A18,INTEGRA1:def 4;
A22: D.i<=upper_bound divset(D1,j) by A19,INTEGRA2:1;
delta(D1) >= min rng upper_volume(chi(A,A),D)
proof
per cases;
suppose
A23: i=1;
len D in Seg len D by FINSEQ_1:3;
then 1 <= len D by FINSEQ_1:1;
then
A24: 1 in dom D by FINSEQ_3:25;
then
A25: lower_bound divset(D,1)=lower_bound A by INTEGRA1:def 4;
1 in Seg len D by A24,FINSEQ_1:def 3;
then 1 in Seg len upper_volume(chi(A,A),D) by INTEGRA1:def 6;
then
A26: 1 in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3;
vol(divset(D,1)) = upper_volume(chi(A,A),D).1 by A24,
INTEGRA1:20;
then vol(divset(D,1)) in rng upper_volume(chi(A,A),D) by A26,
FUNCT_1:def 3;
then
A27: vol
(divset(D,1))>=min rng upper_volume(chi(A,A),D) by XXREAL_2:def 7;
A28: upper_bound divset(D,1)=D. 1 by A24,INTEGRA1:def 4;
upper_bound divset(D1,j)-lower_bound A >= D.1-lower_bound
A by A22,A23,XREAL_1:9;
then
vol(divset(D1,j)) >= upper_bound divset(D,1)- lower_bound
divset(D,1) by A21,A25,A28,INTEGRA1:def 5;
then
A29: vol(divset(D1,j)) >= vol(divset(D,1)) by INTEGRA1:def 5;
vol(divset(D1,j)) <= delta(D1) by A18,Lm5;
then delta(D1) >= vol(divset(D,1)) by A29,XXREAL_0:2;
hence thesis by A27,XXREAL_0:2;
end;
suppose
A30: i<>1;
then D.(i-1) in A by A17,INTEGRA1:7;
then
A31: lower_bound A <= D.(i-1) by INTEGRA2:1;
lower_bound divset(D,i)=D.(i-1) by A17,A30,INTEGRA1:def 4;
then
A32: upper_bound divset(D,i)-lower_bound A >= upper_bound
divset(D,i)-lower_bound divset(D,i) by A31,XREAL_1:10;
upper_bound divset(D,i) = D.i by A17,A30,INTEGRA1:def 4;
then upper_bound divset(D1,j)-lower_bound divset(D1,j) >=
upper_bound divset(D,i)- lower_bound A by A22,A21,XREAL_1:9;
then upper_bound divset(D1,j)-lower_bound divset(D1,j) >=
upper_bound divset(D,i)-lower_bound divset( D,i) by A32,XXREAL_0:2;
then vol(divset(D1,j)) >= upper_bound divset(D,i)-lower_bound
divset(D,i) by INTEGRA1:def 5;
then
A33: vol(divset(D1,j)) >= vol(divset(D,i)) by INTEGRA1:def 5;
i in Seg len D by A17,FINSEQ_1:def 3;
then i in Seg len upper_volume(chi(A,A),D) by INTEGRA1:def 6;
then
A34: i in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3;
vol(divset(D,i)) = upper_volume(chi(A,A),D).i by A17,
INTEGRA1:20;
then vol(divset(D,i)) in rng upper_volume(chi(A,A),D) by A34,
FUNCT_1:def 3;
then
A35: vol
(divset(D,i))>=min rng upper_volume(chi(A,A),D) by XXREAL_2:def 7;
vol(divset(D1,j)) <= delta(D1) by A18,Lm5;
then delta(D1) >= vol(divset(D,i)) by A33,XXREAL_0:2;
hence thesis by A35,XXREAL_0:2;
end;
end;
hence contradiction by A8;
end;
A36: P[1]
proof
len D in Seg len D by FINSEQ_1:3;
then 1 <= len D by FINSEQ_1:1;
then
A37: 1 in dom D by FINSEQ_3:25;
then consider j such that
A38: j in dom D1 and
A39: D.1 in divset(D1,j) by Th3,INTEGRA1:6;
PUf(D1,j)-PUf(D2,indx(D2,D1,j))<= 1*(upper_bound rng f-
lower_bound rng f)*delta(D1)
proof
A40: j <> 1 by A16,A37,A38,A39;
then reconsider j1=j-1 as Element of NAT by A38,INTEGRA1:7;
A41: j1 in dom D1 by A38,A40,INTEGRA1:7;
then j1 in Seg len D1 by FINSEQ_1:def 3;
then j1 in Seg len upper_volume(f,D1) by INTEGRA1:def 6;
then
A42: j1 in dom upper_volume(f,D1) by FINSEQ_1:def 3;
A43: j-1 in dom D1 by A38,A40,INTEGRA1:7;
then
A44: indx(D2,D1,j1) in dom D2 by A10,INTEGRA1:def 19;
then
A45: indx(D2,D1,j1) in Seg len D2 by FINSEQ_1:def 3;
then
A46: 1 <= indx(D2,D1,j1) by FINSEQ_1:1;
then mid(D2,1,indx(D2,D1,j1)) is increasing by A44,INTEGRA1:35;
then
A47: D2|indx(D2,D1,j1) is increasing by A46,FINSEQ_6:116;
j < j+1 by NAT_1:13;
then j1 < j by XREAL_1:19;
then
A48: indx(D2,D1,j1) < indx(D2,D1,j) by A10,A38,A41,Th8;
then
A49: indx(D2,D1,j1)+1 <= indx(D2,D1,j) by NAT_1:13;
A50: Sum mid(upper_volume(f,D1),j,j) -Sum mid(upper_volume(f,D2
),(indx(D2,D1,j1)+1),indx(D2,D1,j)) <= (upper_bound rng f-lower_bound rng f)*
delta(D1)
proof
A51: indx(D2,D1,j)-indx(D2,D1,j1) <= 2
proof
reconsider ID1=indx(D2,D1,j1)+1 as Element of NAT;
reconsider ID2=ID1+1 as Element of NAT;
assume indx(D2,D1,j)-indx(D2,D1,j1) > 2;
then
A52: indx(D2,D1,j1)+(1+1)<indx(D2,D1,j) by XREAL_1:20;
A53: ID1 < ID2 by NAT_1:13;
then indx (D2,D1,j1) <= ID2 by NAT_1:13;
then
A54: 1 <= ID2 by A46,XXREAL_0:2;
A55: indx(D2,D1,j) in dom D2 by A10,A38,INTEGRA1:def 19;
then
A56: indx(D2,D1,j) <= len D2 by FINSEQ_3:25;
then ID2 <= len D2 by A52,XXREAL_0:2;
then
A57: ID2 in dom D2 by A54,FINSEQ_3:25;
then
A58: D2.ID2<D2.indx(D2,D1, j) by A52,A55,SEQM_3:def 1;
A59: 1 <= ID1 by A46,NAT_1:13;
A60: D1.j = D2.indx(D2,D1,j) by A10,A38,INTEGRA1:def 19;
ID1 <= indx(D2,D1,j) by A52,A53,XXREAL_0:2;
then ID1 <= len D2 by A56,XXREAL_0:2;
then
A61: ID1 in dom D2 by A59,FINSEQ_3:25;
then
A62: D2.ID1<D2.ID2 by A53,A57,SEQM_3:def 1;
indx(D2,D1,j1) < ID1 by NAT_1:13;
then
A63: D2 .indx(D2,D1,j1)<D2.ID1 by A44,A61,SEQM_3:def 1;
A64: D1.j1 = D2.indx(D2,D1,j1) by A10,A41,INTEGRA1:def 19;
A65: not D2.ID1 in rng D1 & not D2.ID2 in rng D1
proof
assume
A66: D2.ID1 in rng D1 or D2.ID2 in rng D1;
per cases by A66;
suppose
D2.ID1 in rng D1;
then consider n such that
A67: n in dom D1 and
A68: D1.n=D2.ID1 by PARTFUN1:3;
j1<n by A41,A63,A64,A67,A68,SEQ_4:137;
then
A69: j<n+1 by XREAL_1:19;
D2.ID1<D2.indx(D2,D1,j) by A62,A58,XXREAL_0:2;
then n<j by A38,A60,A67,A68,SEQ_4:137;
hence contradiction by A69,NAT_1:13;
end;
suppose
D2.ID2 in rng D1;
then consider n such that
A70: n in dom D1 and
A71: D1.n=D2.ID2 by PARTFUN1:3;
D2.indx(D2,D1,j1)<D2.ID2 by A63,A62,XXREAL_0:2;
then j1<n by A41,A64,A70,A71,SEQ_4:137;
then
A72: j<n+1 by XREAL_1:19;
n<j by A38,A58,A60,A70,A71,SEQ_4:137;
hence contradiction by A72,NAT_1:13;
end;
end;
upper_bound divset(D1,j)=D1.j by A38,A40,INTEGRA1:def 4;
then
A73: upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A10,A38,
INTEGRA1:def 19;
lower_bound divset(D1,j)=D1.j1 by A38,A40,INTEGRA1:def 4;
then
A74: lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A10,A41,
INTEGRA1:def 19;
D2.ID2 in rng D2 by A57,FUNCT_1:def 3;
then
A75: D2.ID2 in rng D by A11,A65,XBOOLE_0:def 3;
D2 .ID1 in rng D2 by A61,FUNCT_1:def 3;
then
A76: D2.ID1 in rng D by A11,A65,XBOOLE_0:def 3;
D2.indx(D2,D1,j1)<=D2.ID2 by A63,A62,XXREAL_0:2;
then D2.ID2 in divset(D1,j) by A58,A74,A73,INTEGRA2:1;
then
A77: D2.ID2 in rng D /\ divset(D1,j) by A75,XBOOLE_0:def 4;
D2.ID1<=D2.indx(D2,D1,j) by A62,A58,XXREAL_0:2;
then D2.ID1 in divset(D1,j) by A63,A74,A73,INTEGRA2:1;
then D2.ID1 in rng D /\ divset(D1,j) by A76,XBOOLE_0:def 4;
hence contradiction by A8,A38,A53,A61,A57,A77,Th5,SEQ_4:138;
end;
A78: 1 <= indx(D2,D1,j1)+1 by A46,NAT_1:13;
j <= len D1 by A38,FINSEQ_3:25;
then
A79: j <= len upper_volume(f,D1) by INTEGRA1:def 6;
A80: 1 <= j by A38,FINSEQ_3:25;
then
A81: mid(upper_volume(f,D1),j,j).1 = upper_volume(f,D1).j by A79,
FINSEQ_6:118;
reconsider uv = upper_volume(f,D1).j as Element of REAL
by XREAL_0:def 1;
j-'j+1 = 1 by Lm1;
then len mid(upper_volume(f,D1),j,j) = 1 by A80,A79,
FINSEQ_6:118;
then mid(upper_volume(f,D1),j,j) =<*uv*> by A81,FINSEQ_1:40;
then
A82: Sum mid(upper_volume(f,D1),j,j)= upper_volume(f,D1).j by
FINSOP_1:11;
indx(D2,D1,j) in dom D2 by A10,A38,INTEGRA1:def 19;
then
A83: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3;
then
A84: 1 <= indx(D2,D1,j) by FINSEQ_1:1;
indx(D2,D1,j) in Seg len upper_volume(f,D2) by A83,
INTEGRA1:def 6;
then
A85: indx(D2,D1,j) <= len upper_volume(f,D2) by FINSEQ_1:1;
then
A86: indx(D2,D1,j1)+1 <= len upper_volume(f,D2 ) by A49,XXREAL_0:2;
then indx(D2,D1,j1)+1 in Seg len upper_volume(f,D2) by A78,
FINSEQ_1:1;
then
A87: indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 6;
then
A88: indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1) =indx(D2,D1,j)-(indx(
D2,D1,j1)+1) by A49,XREAL_1:233;
then indx(D2,D1,j)-'(indx(D2,D1,j1) +1)+1 <= 2 by A51;
then
A89: len mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j))<=2 by A49,A84,A85,A78,A86,FINSEQ_6:118;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 0+1 by XREAL_1:6;
then
A90: 1 <= len mid(upper_volume(f,D2),(indx(D2, D1,j1)+1),indx
(D2,D1,j )) by A49,A84,A85,A78,A86,FINSEQ_6:118;
now
per cases by A90,A89,Lm2;
suppose
A91: len mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),
indx(D2,D1,j))=1;
upper_bound divset(D1,j)=D1.j by A38,A40,INTEGRA1:def 4;
then
A92: upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A10,A38,
INTEGRA1:def 19;
lower_bound divset(D1,j)=D1.j1 by A38,A40,INTEGRA1:def 4;
then lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A10,A41,
INTEGRA1:def 19;
then
A93: divset(D1,j)=[. D2.indx(D2,D1, j1),D2.indx(D2,D1,j)
.] by A92,INTEGRA1:4;
A94: delta(D1) >= 0 by Th9;
A95: upper_bound rng f - lower_bound rng f >= 0 by A1,Lm3,
XREAL_1:48;
A96: indx(D2,D1,j) in dom D2 by A10,A38,INTEGRA1:def 19;
len mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(
D2,D1,j)) = indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 by A49,A84,A85,A78,A86,
FINSEQ_6:118;
then
A97: indx(D2,D1,j)-(indx(D2,D1,j1)+1)=0 by A49,A91,XREAL_1:233;
then indx(D2,D1,j)<>1 by A45,FINSEQ_1:1;
then
A98: upper_bound divset(D2,indx(D2,D1,j))=D2.indx(D2,D1,j
) by A96,INTEGRA1:def 4;
indx(D2,D1,j)-1=indx(D2,D1,j1) by A97;
then lower_bound divset(D2,indx(D2,D1,j))= D2.indx(D2,D1,
j1) by A46,A97,A96,INTEGRA1:def 4;
then
A99: divset(D2,indx(D2,D1,j))=divset(D1,j) by A93,A98,INTEGRA1:4
;
reconsider uv = upper_volume(f,D2).(indx(D2,D1,j1)+1)
as Element of REAL
by XREAL_0:def 1;
mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)).1 =upper_volume(f,D2).(indx(D2,D1,j1)+1) by A84,A85,A78,A86,FINSEQ_6:118;
then mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)) =<*uv*> by A91,FINSEQ_1:40;
then Sum mid(upper_volume(f,D2),(indx( D2,D1,j1)+1),indx(
D2,D1,j )) =upper_volume(f,D2).(indx(D2,D1,j1)+1) by FINSOP_1:11
.=(upper_bound(rng(f|divset(D2,(indx(D2,D1,j1)+1)))))
*vol(divset(D2,(indx(D2,D1,j1)+1))) by A88,INTEGRA1:def 6
.=Sum mid(upper_volume(f,D1),j,j) by A38,A82,A97,A99,
INTEGRA1:def 6;
hence thesis by A94,A95;
end;
suppose
A100: len mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),
indx(D2,D1,j))=2;
A101: mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)).1 =upper_volume(f,D2).(indx(D2,D1,j1)+1) by A84,A85,A78,A86,FINSEQ_6:118;
A102: 2+(indx(D2,D1,j1)+1)>=0+1 by XREAL_1:7;
mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)).2 =UVf(D2).(2+(indx(D2,D1,j1)+1)-'1) by A49,A84,A85,A78,A86,A100,
FINSEQ_6:118
.=UVf(D2).(2+(indx(D2,D1,j1)+1)-1) by A102,XREAL_1:233
.=UVf(D2).(indx(D2,D1,j1)+(1+1));
then mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)) =<*upper_volume(f,D2).(indx(D2,D1,j1)+1), upper_volume(f,D2).(indx(D2,D1,
j1)+2)*> by A100,A101,FINSEQ_1:44;
then
A103: Sum mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(
D2,D1,j)) =upper_volume(f,D2).(indx(D2,D1,j1)+1) +upper_volume(f,D2).(indx(D2,
D1,j1)+2) by RVSUM_1:77;
A104: vol(divset(D2,indx(D2,D1,j1)+1))>=0 by INTEGRA1:9;
upper_bound divset(D1,j)=D1.j by A38,A40,INTEGRA1:def 4;
then
A105: upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A10,A38,
INTEGRA1:def 19;
A106: vol(divset(D2,indx(D2,D1,j1)+2))>=0 by INTEGRA1:9;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A49,A84,A85,A78
,A86,A100,FINSEQ_6:118;
then
A107: 2=indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1 by A49,XREAL_1:233
.=indx(D2,D1,j)-indx(D2,D1,j1);
then
A108: indx (D2,D1,j1)+2 in dom D2 by A10,A38,INTEGRA1:def 19;
lower_bound divset(D1,j)=D1.j1 by A38,A40,INTEGRA1:def 4;
then lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A10,A41,
INTEGRA1:def 19;
then
A109: vol(divset(D1,j)) =D2.(indx(D2,D1,j1)+2)-D2.(indx(D2
,D1,j1)+1) +D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A105,A107,INTEGRA1:def 5
;
indx(D2,D1,j1)+1 in Seg len upper_volume(f,D2) by A78,A86,
FINSEQ_1:1;
then indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 6;
then
A110: indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3;
A111: indx(D2,D1,j1)+1 <> 1 by A46,NAT_1:13;
then
A112: upper_bound divset(D2,(indx( D2,D1,j1)+1))= D2.(indx
(D2,D1,j1)+1) by A110,INTEGRA1:def 4;
indx(D2,D1,j1)+1-1=indx(D2,D1,j1)+0;
then
A113: lower_bound divset(D2,(indx(D2,D1,j1)+1))= D2.indx(
D2,D1,j1) by A110,A111,INTEGRA1:def 4;
A114: indx(D2,D1,j1)+1+1 > 1 by A78,NAT_1:13;
indx(D2,D1,j1)+2-1=indx(D2,D1,j1)+1;
then
A115: lower_bound divset(D2,(indx(D2,D1,j1)+2))= D2.(indx(
D2,D1,j1)+1) by A108,A114,INTEGRA1:def 4;
upper_bound divset(D2,(indx(D2,D1,j1)+2))= D2.(indx(
D2,D1,j1)+2) by A108,A114,INTEGRA1:def 4;
then vol(divset(D1,j))=vol(divset(D2,indx(D2,D1,j1)+2)) +
D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A115,A109,INTEGRA1:def 5
.=vol(divset(D2,indx(D2,D1,j1)+2)) +(upper_bound
divset(D2,indx(D2,D1,j1)+1)-lower_bound divset(D2, indx(D2,D1,j1)+1)) by A113
,A112;
then
A116: vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+1))+
vol(divset(D2,indx(D2,D1,j1)+2)) by INTEGRA1:def 5;
then
A117: upper_volume(f,D1).j=(upper_bound(rng(f| divset(D1,j
))))* (vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2))) by
A38,INTEGRA1:def 6;
A118: Sum mid(UVf(D1),j,j)-Sum mid(UVf(D2),indx(D2,D1,j1)+
1,indx(D2, D1,j)) <=(upper_bound rng f - lower_bound rng f)* (vol(divset(D2,
indx(D2,D1,j1)+2)) +vol(divset(D2,indx(D2,D1,j1)+1)))
proof
set ID2=indx(D2,D1,j1)+2;
set ID1=indx(D2,D1,j1)+1;
set SR = upper_bound rng(f|divset(D2,ID1)), VR = vol(
divset(D2,ID1));
set B = vol(divset(D2,ID1));
set C = vol(divset(D2,ID2));
divset(D1,j)c=A by A38,INTEGRA1:8;
then
A119: upper_bound rng(f|divset(D1,j)) <= upper_bound rng
f by A1,Lm4;
ID1 in dom D2 by A87,FINSEQ_1:def 3;
then divset(D2,ID1)c=A by INTEGRA1:8;
then upper_bound rng(f|divset(D2,ID1)) >= lower_bound
rng f by A1,Lm4;
then
A120: (upper_bound rng(f|divset(D2,ID1)))* vol(divset(D2
,ID1)) >=(lower_bound rng f)*vol(divset(D2,ID1)) by A104,XREAL_1:64;
indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A49,A84,A85,A78
,A86,A100,FINSEQ_6:118;
then
A121: 2=indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1 by A49,XREAL_1:233
.=indx(D2,D1,j)-indx(D2,D1,j1);
A122: indx(D2,D1,j) in dom D2 by A10,A38,INTEGRA1:def 19;
then divset(D2,ID2) c= A by A121,INTEGRA1:8;
then
A123: upper_bound rng(f|divset(D2,ID2)) >= lower_bound
rng f by A1,Lm4;
reconsider A = upper_bound rng(f|divset(D1,j)) as Real;
A124: upper_volume(f,D1).j-A*B = A*C by A117;
(upper_bound rng(f|divset(D1,j)))*vol( divset(D2,
ID2)) <=(upper_bound rng f)*vol(divset(D2,ID2)) by A106,A119,XREAL_1:64;
then Sum mid(UVf(D1),j,j)<=(upper_bound rng(f|divset(D1
,j)))* vol(divset(D2, ID1)) +(upper_bound rng f)*vol(divset(D2,ID2)) by A82
,A124,XREAL_1:20;
then
A125: Sum mid(UVf(D1),j,j)-(upper_bound rng f)* vol(
divset(D2,ID2) ) <=(upper_bound rng(f|divset(D1,j)))*vol(divset(D2,ID1)) by
XREAL_1:20;
(upper_bound rng(f|divset(D1,j)))*vol(divset(D2,
ID1)) <=(upper_bound rng f)*vol(divset(D2,ID1)) by A104,A119,XREAL_1:64;
then Sum mid(UVf(D1),j,j)-(upper_bound rng f)* vol(
divset(D2,ID2) ) <=(upper_bound rng f)*vol(divset(D2,ID1)) by A125,XXREAL_0:2;
then
A126: Sum mid(UVf(D1),j,j) <=(upper_bound rng f)*vol(
divset(D2,ID2))+(upper_bound rng f)* vol(divset(D2,ID1)) by XREAL_1:20;
Sum mid(UVf(D2),ID1,indx(D2,D1,j)) =(upper_bound
rng(f|divset(D2,ID2)))*vol(divset(D2,ID2)) +UVf(D2).ID1 by A103,A122,A121,
INTEGRA1:def 6
.=(upper_bound rng(f|divset(D2,ID2)))* vol(divset(D2
,ID2)) +(upper_bound rng(f|divset(D2,ID1))) *vol(divset(D2,ID1)) by A88,
INTEGRA1:def 6;
then Sum mid(UVf(D2),ID1,indx(D2,D1,j)) - SR*VR >=(
lower_bound rng f)*vol(divset(D2,ID2)) by A106,A123,XREAL_1:64;
then Sum mid(UVf(D2),ID1,indx(D2,D1,j)) >=(lower_bound
rng f)*vol(divset(D2,ID2))+SR*VR by XREAL_1:19;
then Sum mid(UVf(D2),ID1,indx(D2,D1,j))-(lower_bound
rng f)* vol(divset(D2, ID2)) >=SR*VR by XREAL_1:19;
then Sum mid(UVf(D2),ID1,indx(D2,D1,j))-(lower_bound
rng f)* vol(divset(D2, ID2)) >=(lower_bound rng f)*VR by A120,XXREAL_0:2;
then Sum mid(UVf(D2),ID1,indx(D2,D1,j))>=(lower_bound
rng f)*vol(divset(D2, ID2))+ (lower_bound rng f)*VR by XREAL_1:19;
then Sum mid(UVf(D1),j,j)-Sum mid(UVf(D2),ID1,indx(D2,
D1,j)) <=(upper_bound rng f)*vol(divset(D2,ID2))+(upper_bound rng f)* vol(
divset(D2,ID1)) -((lower_bound rng f)*vol(divset(D2,ID2))+(lower_bound rng f)*
vol(divset(D2,ID1))) by A126,XREAL_1:13;
hence thesis;
end;
upper_bound rng f - lower_bound rng f >= 0 by A1,Lm3,
XREAL_1:48;
then (upper_bound rng f - lower_bound rng f)*( vol(divset
(D1,j))) <=(upper_bound rng f - lower_bound rng f)*delta(D1) by A38,Lm5,
XREAL_1:64;
hence thesis by A116,A118,XXREAL_0:2;
end;
end;
hence thesis;
end;
j < j+1 by NAT_1:13;
then
A127: j1 < j by XREAL_1:19;
indx(D2,D1,j) in dom D2 by A10,A38,INTEGRA1:def 19;
then
A128: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3;
then
A129: 1 <= indx(D2,D1,j) by FINSEQ_1:1;
A130: indx(D2,D1,j1) <= len D2 by A45,FINSEQ_1:1;
then
A131: len (D2|indx(D2,D1,j1))=indx(D2,D1,j1) by FINSEQ_1:59;
A132: j1 in Seg len D1 by A43,FINSEQ_1:def 3;
then
A133: j1 <= len D1 by FINSEQ_1:1;
for x1 being object st x1 in rng(D1|j1)
holds x1 in rng(D2|indx(D2,D1,j1))
proof
let x1 be object;
assume x1 in rng(D1|j1);
then consider k such that
A134: k in dom(D1|j1) and
A135: x1=(D1|j1).k by PARTFUN1:3;
k in Seg len(D1|j1) by A134,FINSEQ_1:def 3;
then
A136: k in Seg j1 by A133,FINSEQ_1:59;
then
A137: k in dom D1 by A41,RFINSEQ:6;
k <= j1 by A136,FINSEQ_1:1;
then D1.k <= D1.j1 by A43,A137,SEQ_4:137;
then D2.indx(D2,D1,k) <= D1.j1 by A10,A137,INTEGRA1:def 19;
then
A138: D2.indx(D2,D1,k)<=D2.indx(D2,D1,j1) by A10,A43,INTEGRA1:def 19;
A139: (D1|j1).k = D1.k by A41,A136,RFINSEQ:6;
D1.k in rng D1 by A137,FUNCT_1:def 3;
then x1 in rng D2 by A11,A135,A139,XBOOLE_0:def 3;
then consider n such that
A140: n in dom D2 and
A141: x1=D2.n by PARTFUN1:3;
D2.indx(D2,D1,k)=D2.n by A10,A135,A139,A137,A141,
INTEGRA1:def 19;
then
A142: n <= indx(D2,D1,j1) by A44,A140,A138,SEQM_3:def 1;
1 <= n by A140,FINSEQ_3:25;
then
A143: n in Seg indx(D2,D1,j1) by A142,FINSEQ_1:1;
then n in Seg len(D2|indx(D2,D1,j1)) by A130,FINSEQ_1:59;
then
A144: n in dom(D2|indx(D2,D1,j1)) by FINSEQ_1:def 3;
D2.n = (D2|indx(D2,D1,j1)).n by A44,A143,RFINSEQ:6;
hence thesis by A141,A144,FUNCT_1:def 3;
end;
then
A145: rng(D1|j1) c= rng(D2|indx(D2,D1,j1));
A146: 1 <= j1 by A132,FINSEQ_1:1;
lower_bound divset(D1,j) <= D.1 by A39,INTEGRA2:1;
then
A147: D1.j1 <= D.1 by A38,A40,INTEGRA1:def 4;
for x1 being object st x1 in rng(D2|indx(D2,D1,j1))
holds x1 in rng(D1|j1)
proof
let x1 be object;
assume x1 in rng(D2|indx(D2,D1,j1));
then consider k such that
A148: k in dom(D2|indx(D2,D1,j1)) and
A149: x1=(D2|indx(D2,D1,j1)).k by PARTFUN1:3;
k in Seg len(D2|indx(D2,D1,j1)) by A148,FINSEQ_1:def 3;
then
A150: k in Seg indx(D2,D1,j1) by A130,FINSEQ_1:59;
then
A151: k in dom D2 by A44,RFINSEQ:6;
A152: len(D1|j1) = j1 by A133,FINSEQ_1:59;
k <= indx(D2,D1,j1) by A150,FINSEQ_1:1;
then D2.k <= D2.indx(D2,D1,j1) by A44,A151,SEQ_4:137;
then
A153: D2.k <= D1.j1 by A10,A43,INTEGRA1:def 19;
A154: D2.k in rng D1 implies D2.k in rng(D1|j1)
proof
assume D2.k in rng D1;
then consider m such that
A155: m in dom D1 and
A156: D2.k = D1.m by PARTFUN1:3;
m in Seg len D1 by A155,FINSEQ_1:def 3;
then
A157: 1 <= m by FINSEQ_1:1;
A158: m <= j1 by A41,A153,A155,A156,SEQM_3:def 1;
then m in Seg len(D1|j1) by A152,A157,FINSEQ_1:1;
then
A159: m in dom (D1|j1) by FINSEQ_1:def 3;
m in Seg j1 by A157,A158,FINSEQ_1:1;
then D2.k = (D1|j1).m by A41,A156,RFINSEQ:6;
hence thesis by A159,FUNCT_1:def 3;
end;
A160: D2.k in rng D implies D2.k = D1.j1
proof
assume D2.k in rng D;
then consider n such that
A161: n in dom D and
A162: D2.k=D.n by PARTFUN1:3;
1 <= n by A161,FINSEQ_3:25;
then D.1 <= D2.k by A37,A161,A162,SEQ_4:137;
then D1.j1 <= D2.k by A147,XXREAL_0:2;
hence thesis by A153,XXREAL_0:1;
end;
A163: D2.k in rng D implies D2.k in rng(D1|j1)
proof
j1 in Seg len(D1|j1) by A146,A152,FINSEQ_1:1;
then j1 in dom(D1|j1) by FINSEQ_1:def 3;
then
A164: (D1|j1).j1 in rng(D1|j1) by FUNCT_1:def 3;
assume
A165: D2.k in rng D;
j1 in Seg j1 by A146,FINSEQ_1:1;
hence thesis by A41,A160,A165,A164,RFINSEQ:6;
end;
D2.k in rng D2 by A151,FUNCT_1:def 3;
hence thesis by A11,A44,A149,A150,A163,A154,RFINSEQ:6
,XBOOLE_0:def 3;
end;
then rng(D2|indx(D2,D1,j1)) c= rng (D1|j1);
then
A166: rng (D2|indx(D2,D1,j1)) = rng (D1|j1) by A145,XBOOLE_0:def 10;
mid(D1,1,j1) is increasing by A38,A40,A146,INTEGRA1:7,35;
then
A167: D1|j1 is increasing by A146,FINSEQ_6:116;
then
A168: D2|indx(D2,D1,j1)=D1|j1 by A47,A166,Th6;
A169: for k st 1 <= k & k <= j1 holds k=indx(D2,D1,k)
proof
let k;
assume that
A170: 1 <= k and
A171: k <= j1;
assume
A172: k<>indx(D2,D1,k);
now
per cases by A172,XXREAL_0:1;
suppose
A173: k > indx(D2,D1,k);
k <= len D1 by A133,A171,XXREAL_0:2;
then
A174: k in dom D1 by A170,FINSEQ_3:25;
then indx(D2,D1,k) in dom D2 by A10,INTEGRA1:def 19;
then indx(D2,D1,k) in Seg len D2 by FINSEQ_1:def 3;
then
A175: 1<=indx(D2,D1,k) by FINSEQ_1:1;
A176: indx(D2,D1,k) < j1 by A171,A173,XXREAL_0:2;
then
A177: indx(D2,D1,k) in Seg j1 by A175,FINSEQ_1:1;
indx(D2,D1,k)<= indx(D2,D1,j1) by A10,A41,A171,A174,Th7;
then indx(D2,D1,k) in Seg indx(D2,D1,j1) by A175,FINSEQ_1:1
;
then
A178: (D2|indx(D2,D1,j1)).indx(D2,D1,k)= D2.indx(D2,D1,k)
by A44,RFINSEQ:6;
indx(D2,D1,k) <= len D1 by A133,A176,XXREAL_0:2;
then indx(D2,D1,k) in dom D1 by A175,FINSEQ_3:25;
then
A179: D1.k > D1.indx(D2,D1,k) by A173,A174,SEQM_3:def 1;
D1.k=D2. indx(D2,D1,k) by A10,A174,INTEGRA1:def 19;
hence contradiction by A41,A168,A178,A179,A177,RFINSEQ:6;
end;
suppose
A180: k < indx(D2,D1,k);
k <= len D1 by A133,A171,XXREAL_0:2;
then
A181: k in dom D1 by A170,FINSEQ_3:25;
then indx(D2,D1,k) <= indx(D2,D1,j1) by A10,A41,A171,Th7;
then
A182: k <= indx(D2,D1,j1) by A180,XXREAL_0:2;
then k <= len D2 by A130,XXREAL_0:2;
then
A183: k in dom D2 by A170,FINSEQ_3:25;
k in Seg j1 by A170,A171,FINSEQ_1:1;
then
A184: D1.k = (D1|j1).k by A43,RFINSEQ:6;
indx(D2,D1,k) in dom D2 by A10,A181,INTEGRA1:def 19;
then
A185: D2.k < D2.indx(D2,D1,k) by A180,A183,SEQM_3:def 1;
A186: k in Seg indx(D2,D1,j1) by A170,A182,FINSEQ_1:1;
D1.k=D2. indx(D2,D1,k) by A10,A181,INTEGRA1:def 19;
hence contradiction by A44,A168,A184,A185,A186,RFINSEQ:6;
end;
end;
hence contradiction;
end;
A187: for k be Nat st 1 <= k & k <= len(upper_volume(f,D1)|j1)
holds (upper_volume(f,D1)|j1).k = (upper_volume(f,D2)|indx(D2,D1,j1)).k
proof
indx(D2,D1,j1) in Seg len D2 by A44,FINSEQ_1:def 3;
then indx(D2,D1,j1) in Seg len upper_volume(f,D2) by
INTEGRA1:def 6;
then
A188: indx(D2,D1,j1) in dom upper_volume(f,D2) by FINSEQ_1:def 3;
let k be Nat;
assume that
A189: 1 <= k and
A190: k <= len(upper_volume(f,D1)|j1);
A191: len(upper_volume(f,D1)) = len D1 by INTEGRA1:def 6;
then
A192: k <= j1 by A133,A190,FINSEQ_1:59;
then
A193: k in Seg j1 by A189,FINSEQ_1:1;
then indx(D2,D1,k) in Seg j1 by A169,A189,A192;
then
A194: indx(D2,D1,k) in Seg indx(D2,D1,j1) by A146,A169;
then indx(D2,D1,k)<=indx(D2,D1,j1) by FINSEQ_1:1;
then
A195: indx(D2,D1,k)<=len D2 by A130,XXREAL_0:2;
k <= len D1 by A133,A192,XXREAL_0:2;
then
A196: k in Seg len D1 by A189,FINSEQ_1:1;
then
A197: k in dom D1 by FINSEQ_1:def 3;
then
A198: indx(D2,D1,k) in dom D2 by A10,INTEGRA1:def 19;
A199: D1.k=D2.indx (D2,D1,k) by A10,A197,INTEGRA1:def 19;
A200: lower_bound divset(D1,k)=lower_bound divset(D2, indx(D2,
D1,k)) & upper_bound divset(D1,k)=upper_bound divset(D2,indx(D2,D1,k))
proof
per cases;
suppose
A201: k=1;
then
A202: upper_bound divset(D1,k)=D1.k by A197,INTEGRA1:def 4;
A203: lower_bound divset(D1,k)=lower_bound A by A197,A201,
INTEGRA1:def 4;
indx(D2,D1,k)=1 by A146,A169,A201;
hence thesis by A198,A199,A203,A202,INTEGRA1:def 4;
end;
suppose
A204: k<>1;
then reconsider k1=k-1 as Element of NAT by A197,INTEGRA1:7
;
k <= k+1 by NAT_1:11;
then k1 <= k by XREAL_1:20;
then
A205: k1 <= j1 by A192,XXREAL_0:2;
A206: k-1 in dom D1 by A197,A204,INTEGRA1:7;
then 1 <= k1 by FINSEQ_3:25;
then k1=indx(D2,D1,k1) by A169,A205;
then
A207: D2.(indx(D2,D1,k)-1)= D2.indx(D2,D1,k1) by A169,A189,A192
,A193;
A208: indx(D2,D1,k)<>1 by A169,A189,A192,A193,A204;
then
A209: lower_bound divset(D2,indx(D2,D1,k))= D2.(indx(D2,D1
,k)-1) by A198,INTEGRA1:def 4;
A210: upper_bound divset(D2,indx(D2,D1,k))=D2.indx(D2,D1,k
) by A198,A208,INTEGRA1:def 4;
A211: upper_bound divset(D1,k)=D1.k by A197,A204,INTEGRA1:def 4;
lower_bound divset(D1,k)=D1.(k-1) by A197,A204,
INTEGRA1:def 4;
hence thesis by A10,A197,A211,A206,A209,A210,A207,
INTEGRA1:def 19;
end;
end;
divset(D2,indx(D2,D1,k))= [. lower_bound divset(D2, indx
(D2,D1,k)), upper_bound divset(D2,indx(D2,D1,k)).] by INTEGRA1:4;
then
A212: divset(D1,k)=divset(D2,indx(D2,D1,k)) by A200,INTEGRA1:4;
A213: k in dom D1 by A196,FINSEQ_1:def 3;
j1 in Seg len(upper_volume(f,D1)) by A41,A191,FINSEQ_1:def 3;
then j1 in dom(upper_volume(f,D1)) by FINSEQ_1:def 3;
then
A214: (upper_volume(f,D1)|j1).k = upper_volume(f,D1).k by A193,
RFINSEQ:6
.=(upper_bound(rng(f|divset(D2,indx(D2,D1,k)))))*vol(
divset(D2,indx(D2,D1,k))) by A213,A212,INTEGRA1:def 6;
1<=indx(D2,D1,k) by A169,A189,A192,A193;
then indx(D2,D1,k) in Seg len D2 by A195,FINSEQ_1:1;
then
A215: indx(D2,D1,k) in dom D2 by FINSEQ_1:def 3;
(upper_volume(f,D2)|indx(D2,D1,j1)).k =(upper_volume(f,
D2)|indx(D2,D1,j1)).indx(D2,D1,k) by A169,A189,A192,A193
.=upper_volume(f,D2).indx(D2,D1,k) by A194,A188,RFINSEQ:6
.=(upper_bound(rng(f|divset(D2,indx(D2,D1,k)))))*vol(
divset(D2,indx(D2,D1,k))) by A215,INTEGRA1:def 6;
hence thesis by A214;
end;
indx(D2,D1,j1) in dom D2 by A10,A43,INTEGRA1:def 19;
then indx(D2,D1,j1) <= len D2 by FINSEQ_3:25;
then
A216: indx(D2,D1,j1) <= len upper_volume(f,D2) by INTEGRA1:def 6;
j1 <= len D1 by A43,FINSEQ_3:25;
then
A217: j1 <= len upper_volume(f,D1) by INTEGRA1:def 6;
len (D2|indx(D2,D1,j1))=len (D1|j1) by A47,A167,A166,Th6;
then indx(D2,D1,j1) =j1 by A133,A131,FINSEQ_1:59;
then len(upper_volume(f,D1)|j1)=indx(D2,D1,j1) by A217,
FINSEQ_1:59;
then len(upper_volume(f,D1)|j1)=len(upper_volume(f, D2)|indx(D2
,D1,j1)) by A216,FINSEQ_1:59;
then
A218: upper_volume(f,D2)|indx(D2,D1,j1)=upper_volume (f,D1)|j1
by A187,FINSEQ_1:14;
A219: j in Seg len D1 by A38,FINSEQ_1:def 3;
then
A220: 1 <= j by FINSEQ_1:1;
indx(D2,D1,j) in Seg len UVf(D2) by A128,INTEGRA1:def 6;
then
A221: indx(D2,D1,j) in dom UVf(D2) by FINSEQ_1:def 3;
indx(D2,D1,j) <= len D2 by A128,FINSEQ_1:1;
then
A222: indx(D2,D1,j) <= len UVf(D2) by INTEGRA1:def 6;
j in Seg len UVf(D1) by A219,INTEGRA1:def 6;
then
A223: j in dom UVf(D1) by FINSEQ_1:def 3;
j <= len D1 by A219,FINSEQ_1:1;
then
A224: j <= len UVf(D1) by INTEGRA1:def 6;
j1 in Seg len D1 by A41,FINSEQ_1:def 3;
then j1 in Seg len UVf(D1) by INTEGRA1:def 6;
then j1 in dom UVf(D1) by FINSEQ_1:def 3;
then PUf(D1,j1)=Sum(UVf(D1)|j1) by INTEGRA1:def 20;
then PUf(D1,j1)+Sum mid(UVf(D1),j,j) =Sum((UVf(D1)|j1)^mid(UVf(
D1),j,j)) by RVSUM_1:75
.=Sum(mid(UVf(D1),1,j1)^mid(UVf(D1),j1+1,j)) by A146,
FINSEQ_6:116
.=Sum(mid(UVf(D1),1,j)) by A146,A224,A127,INTEGRA2:4
.=Sum(UVf(D1)|j) by A220,FINSEQ_6:116;
then
A225: PUf(D1,j1)+Sum mid(upper_volume(f,D1),j,j)=PUf(D1,j) by A223,
INTEGRA1:def 20;
indx(D2,D1,j1) in Seg len D2 by A44,FINSEQ_1:def 3;
then indx(D2,D1,j1) in Seg len UVf(D2) by INTEGRA1:def 6;
then indx(D2,D1,j1) in dom UVf(D2) by FINSEQ_1:def 3;
then PUf(D2,indx(D2,D1,j1)) =Sum(UVf(D2)|indx(D2,D1,j1)) by
INTEGRA1:def 20;
then PUf(D2,indx(D2,D1,j1)) +Sum mid(upper_volume(f,D2),(indx(
D2,D1,j1)+1),indx(D2,D1,j)) =Sum(UVf(D2)|indx(D2,D1,j1) ^mid(UVf(D2),(indx(D2,
D1,j1)+1),indx(D2,D1,j))) by RVSUM_1:75
.=Sum(mid(UVf(D2),1,indx(D2,D1,j1)) ^mid(UVf(D2),indx(D2,D1,
j1)+1,indx(D2,D1,j))) by A46,FINSEQ_6:116
.=Sum(mid(UVf(D2),1,indx(D2,D1,j))) by A46,A48,A222,INTEGRA2:4
.=Sum(UVf(D2)|indx(D2,D1,j)) by A129,FINSEQ_6:116;
then
A226: PUf(D2,indx(D2,D1,j1)) +Sum mid(upper_volume(f,D2),(indx(
D2,D1,j1)+1), indx(D2,D1,j)) =PUf(D2,indx(D2,D1,j)) by A221,INTEGRA1:def 20;
indx(D2,D1,j1) in Seg len D2 by A44,FINSEQ_1:def 3;
then indx(D2,D1,j1) in Seg len upper_volume(f,D2) by
INTEGRA1:def 6;
then indx(D2,D1,j1) in dom upper_volume(f,D2) by FINSEQ_1:def 3;
then PUf(D2,indx(D2,D1,j1))=Sum(upper_volume(f,D2)|indx(D2,D1,
j1)) by INTEGRA1:def 20
.=PUf(D1,j1) by A218,A42,INTEGRA1:def 20;
hence thesis by A50,A226,A225;
end;
hence thesis by A38,A39;
end;
reconsider i as non zero Element of NAT by A15,FINSEQ_1:1;
A227: for i being non zero Nat st P[i] holds P[i+1]
proof
let i be non zero Nat;
A228: i>=1 by NAT_1:14;
assume
A229: P[i];
P[i+1]
proof
A230: i <= i+1 by NAT_1:11;
assume
A231: i+1 in dom D;
then consider j such that
A232: j in dom D1 and
A233: D.(i+1) in divset(D1,j) by Th3,INTEGRA1:6;
A234: D2.indx(D2,D1,j)=D1.j by A10,A232,INTEGRA1:def 19;
i+1 in Seg len D by A231,FINSEQ_1:def 3;
then i+1 <= len D by FINSEQ_1:1;
then i <= len D by A230,XXREAL_0:2;
then
A235: i in Seg len D by A228,FINSEQ_1:1;
then
A236: i in dom D by FINSEQ_1:def 3;
A237: indx(D2,D1,j) in dom D2 by A10,A232,INTEGRA1:def 19;
then
A238: 1 <= indx(D2,D1,j) by FINSEQ_3:25;
A239: indx(D2,D1,j) <= len D2 by A237,FINSEQ_3:25;
then
A240: indx(D2,D1,j) <= len UVf(D2) by INTEGRA1:def 6;
consider n1 being Element of NAT such that
A241: n1 in dom D1 and
A242: D.i in divset(D1,n1) and
A243: PUf(D1,n1)-PUf(D2,indx(D2,D1,n1))<= i*(upper_bound rng
f- lower_bound rng f)*delta(D1) by A229,A235,FINSEQ_1:def 3;
A244: 1 <= n1+1 by NAT_1:12;
A245: n1 < j
proof
assume
A246: n1 >= j;
now
per cases by A246,XXREAL_0:1;
suppose
A247: n1=j;
D.i in rng D by A236,FUNCT_1:def 3;
then
A248: D.i in rng D /\ divset(D1,j) by A242,A247,XBOOLE_0:def 4;
D.(i+1) in rng D by A231,FUNCT_1:def 3;
then
A249: D.(i+1) in rng D /\ divset(D1,j) by A233,XBOOLE_0:def 4;
i+1 > i by XREAL_1:29;
hence contradiction by A8,A231,A232,A236,A248,A249,Th5,
SEQ_4:138;
end;
suppose
n1>j;
then
A250: n1>=j+1 by NAT_1:13;
then
A251: n1-1 >= j by XREAL_1:19;
1 <= j by A232,FINSEQ_3:25;
then 1+1 <= j+1 by XREAL_1:6;
then
A252: n1 <> 1 by A250,XXREAL_0:2;
then n1-1 in dom D1 by A241,INTEGRA1:7;
then
A253: D1.j <= D1.(n1-1) by A232,A251,SEQ_4:137;
A254: upper_bound divset(D1,j)=D1.j
proof
per cases;
suppose
j=1;
hence thesis by A232,INTEGRA1:def 4;
end;
suppose
j<>1;
hence thesis by A232,INTEGRA1:def 4;
end;
end;
A255: lower_bound divset(D1,n1) <= D.i by A242,INTEGRA2:1;
lower_bound divset(D1,n1)=D1.(n1-1) by A241,A252,
INTEGRA1:def 4;
then
A256: D.i >= D1.j by A255,A253,XXREAL_0:2;
A257: i < i+1 by XREAL_1:29;
D.(i+1)<=upper_bound divset(D1,j) by A233,INTEGRA2:1;
then D.i >= D.(i+1) by A254,A256,XXREAL_0:2;
hence contradiction by A231,A236,A257,SEQM_3:def 1;
end;
end;
hence thesis;
end;
then
A258: n1+1 <= j by NAT_1:13;
A259: 1 <= n1 by A241,FINSEQ_3:25;
A260: D2.indx(D2,D1,n1) = D1.n1 by A10,A241,INTEGRA1:def 19;
A261: 1 <= j by A232,FINSEQ_3:25;
A262: indx(D2,D1,n1) in dom D2 by A10,A241,INTEGRA1:def 19;
then
A263: 1 <= indx(D2,D1,n1) by FINSEQ_3:25;
A264: j <= len D1 by A232,FINSEQ_3:25;
then
A265: n1+1 <= len D1 by A258,XXREAL_0:2;
then
A266: n1+1 in dom D1 by A244,FINSEQ_3:25;
then
A267: D2.indx(D2,D1,n1+ 1)=D1.(n1+1) by A10,INTEGRA1:def 19;
A268: j <= len UVf(D1) by A264,INTEGRA1:def 6;
then j in Seg len UVf(D1) by A261,FINSEQ_1:1;
then
A269: j in dom UVf(D1) by FINSEQ_1:def 3;
A270: indx(D2,D1,n1+1) in dom D2 by A10,A266,INTEGRA1:def 19;
then
A271: 1 <= indx(D2,D1,n1+1) by FINSEQ_3:25;
n1 in Seg len D1 by A241,FINSEQ_1:def 3;
then n1 in Seg len UVf(D1) by INTEGRA1:def 6;
then n1 in dom UVf(D1) by FINSEQ_1:def 3;
then PUf(D1,n1)=Sum(UVf(D1)|n1) by INTEGRA1:def 20
.=Sum mid(UVf(D1),1,n1) by A259,FINSEQ_6:116;
then PUf(D1,n1)+Sum mid(UVf(D1),n1+1,j) =Sum(mid(UVf(D1),1,n1)
^mid(UVf(D1),n1+1,j)) by RVSUM_1:75
.=Sum mid(UVf(D1),1,j) by A245,A259,A268,INTEGRA2:4
.=Sum(UVf(D1)|j) by A261,FINSEQ_6:116;
then
A272: PUf(D1,j)=PUf(D1,n1)+Sum mid(UVf(D1),n1+1,j) by A269,
INTEGRA1:def 20;
indx(D2,D1,j) in Seg len D2 by A237,FINSEQ_1:def 3;
then indx(D2,D1,j) in Seg len UVf(D2) by INTEGRA1:def 6;
then
A273: indx(D2,D1,j) in dom UVf(D2) by FINSEQ_1:def 3;
A274: indx(D2,D1,n1+1) <= len D2 by A270,FINSEQ_3:25;
D1.(n1+1) <= D1.j by A232,A258,A266,SEQ_4:137;
then
A275: indx(D2,D1,n1+1) <= indx(D2,D1,j) by A270,A267,A237,A234,
SEQM_3:def 1;
then 1+indx(D2,D1,n1+1) <= indx(D2,D1,j)+1 by XREAL_1:6;
then 1 <= indx(D2,D1,j)+1-indx(D2,D1,n1+1) by XREAL_1:19;
then
A276: mid(D2,indx(D2,D1,n1+1),indx(D2,D1,j)).1 =D2.(1-1+indx(D2,
D1,n1+1)) by A275,A271,A239,FINSEQ_6:122
.=D1.(n1+1) by A10,A266,INTEGRA1:def 19;
A277: n1 >= 1 by A241,FINSEQ_3:25;
A278: j-n1 >= 1 by A258,XREAL_1:19;
Sum mid(UVf(D1),n1+1,j)-Sum mid(UVf(D2),indx(D2,D1,n1)+1,
indx(D2,D1,j)) <= (upper_bound rng f-lower_bound rng f)*delta(D1)
proof
per cases by A258,XXREAL_0:1;
suppose
A279: n1+1=j;
A280: indx(D2,D1,j)-indx(D2,D1,n1)<=2
proof
A281: upper_bound divset(D1,j) = D1.j by A232,A245,A277,
INTEGRA1:def 4;
A282: lower_bound divset(D1,j) = D1.(j-1) by A232,A245,A277,
INTEGRA1:def 4;
A283: 1 <= indx(D2,D1,n1)+1 by A263,NAT_1:13;
assume indx(D2,D1,j) - indx(D2,D1,n1) > 2;
then
A284: indx(D2,D1,n1)+2 < indx(D2,D1,j) by XREAL_1:20;
then
A285: indx(D2,D1,n1)+2 <= len D2 by A239,XXREAL_0:2;
A286: indx(D2,D1,n1)+1 < indx(D2,D1,n1)+2 by XREAL_1:6;
then
A287: indx(D2,D1,n1) < indx(D2,D1,n1)+2 by NAT_1:13;
then 1 <= indx(D2,D1,n1)+2 by A263,XXREAL_0:2;
then
A288: indx(D2,D1,n1)+2 in dom D2 by A285,FINSEQ_3:25;
then
A289: D2.indx(D2,D1,j) >= D2.(indx(D2,D1,n1)+2) by A237,A284,
SEQ_4:137;
A290: not D2.(indx(D2,D1,n1)+2) in rng D1
proof
assume D2.(indx(D2,D1,n1)+2) in rng D1;
then consider k1 being Element of NAT such that
A291: k1 in dom D1 and
A292: D2.(indx(D2,D1,n1)+2) = D1.k1 by PARTFUN1:3;
D2.(indx(D2,D1,n1)+2) < D2.indx(D2,D1,j) by A237,A284
,A288,SEQM_3:def 1;
then
A293: k1 < j by A232,A234,A291,A292,SEQ_4:137;
D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+2) by A262,A287
,A288,SEQM_3:def 1;
then n1 < k1 by A241,A260,A291,A292,SEQ_4:137;
hence contradiction by A279,A293,NAT_1:13;
end;
D2.(indx(D2,D1,n1)+2) in rng D2 by A288,FUNCT_1:def 3;
then
A294: D2.(indx(D2,D1,n1)+2) in rng D by A11,A290,XBOOLE_0:def 3;
A295: lower_bound divset(D1,j) = D1.(j-1) by A232,A245,A277,
INTEGRA1:def 4;
A296: upper_bound divset(D1,j) = D1.j by A232,A245,A277,
INTEGRA1:def 4;
D2.(indx(D2,D1,n1)+2) >= D2.indx(D2,D1,n1) by A262,A287
,A288,SEQ_4:137;
then D2.(indx(D2,D1,n1)+2) in divset(D1,j) by A260,A234
,A279,A295,A281,A289,INTEGRA2:1;
then
A297: D2.(indx(D2,D1,n1)+2) in rng D /\ divset(D1,j) by A294,
XBOOLE_0:def 4;
A298: indx(D2,D1,n1)+1 < indx(D2,D1,j) by A284,A286,XXREAL_0:2;
then indx(D2,D1,n1)+1 <= len D2 by A239,XXREAL_0:2;
then
A299: indx(D2,D1,n1)+1 in dom D2 by A283,FINSEQ_3:25;
then
A300: D2.indx(D2,D1,j) >= D2.(indx(D2,D1,n1)+1) by A237,A298,
SEQ_4:137;
A301: indx(D2,D1,n1) < indx(D2,D1,n1)+1 by NAT_1:13;
A302: not D2.(indx(D2,D1,n1)+1) in rng D1
proof
assume D2.(indx(D2,D1,n1)+1) in rng D1;
then consider k1 being Element of NAT such that
A303: k1 in dom D1 and
A304: D2.(indx(D2,D1,n1)+1) = D1.k1 by PARTFUN1:3;
D2.(indx(D2,D1,n1)+1) < D2.indx(D2,D1,j) by A237,A298
,A299,SEQM_3:def 1;
then
A305: k1 < j by A232,A234,A303,A304,SEQ_4:137;
D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+1) by A262,A301
,A299,SEQM_3:def 1;
then n1 < k1 by A241,A260,A303,A304,SEQ_4:137;
hence contradiction by A279,A305,NAT_1:13;
end;
D2.(indx(D2,D1,n1)+1) in rng D2 by A299,FUNCT_1:def 3;
then
A306: D2.(indx(D2,D1,n1)+1) in rng D by A11,A302,XBOOLE_0:def 3;
D2.(indx(D2,D1,n1)+1) >= D2.indx(D2,D1,n1) by A262,A301
,A299,SEQ_4:137;
then D2.(indx(D2,D1,n1)+1) in divset(D1,j) by A260,A234
,A279,A282,A296,A300,INTEGRA2:1;
then D2.(indx(D2,D1,n1)+1) in rng D /\ divset(D1,j) by A306
,XBOOLE_0:def 4;
then D2.(indx(D2,D1,n1)+1) = D2.(indx(D2,D1,n1)+2) by A8
,A232,A297,Th5;
hence contradiction by A286,A299,A288,SEQM_3:def 1;
end;
A307: indx(D2,D1,n1)+1<indx(D2,D1,j) implies indx(D2,D1,n1)+
2=indx(D2, D1, j )
proof
assume indx(D2,D1,n1)+1 < indx(D2,D1,j);
then
A308: indx(D2,D1,n1)+1+1 <= indx(D2, D1,j) by NAT_1:13;
indx(D2,D1,n1)+2 >= indx(D2,D1,j) by A280,XREAL_1:20;
hence thesis by A308,XXREAL_0:1;
end;
A309: 1<=indx(D2,D1,n1)+1 by NAT_1:12;
A310: indx(D2,D1,j) <= len UVf(D2) by A239,INTEGRA1:def 6;
D1.n1 < D1.j by A232,A241,A245,SEQM_3:def 1;
then
A311: indx(D2,D1,n1)<indx(D2,D1,j) by A262,A260,A237,A234,SEQ_4:137
;
then
A312: indx(D2,D1,n1)+1 <= indx(D2,D1,j) by NAT_1:13;
then indx(D2,D1,n1)+1 <= len D2 by A239,XXREAL_0:2;
then indx(D2,D1,n1)+1 <= len UVf(D2) by INTEGRA1:def 6;
then
A313: len mid(UVf(D2),indx(D2,D1,n1)+1,indx( D2,D1,j)) =indx
(D2,D1,j)-'(indx(D2,D1,n1)+1)+1 by A238,A312,A309,A310,FINSEQ_6:118
.=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A312,XREAL_1:233
.=indx(D2,D1,j)-indx(D2,D1,n1);
indx(D2,D1,n1)+1 <= indx(D2,D1,j) by A311,NAT_1:13;
then
A314: indx(D2,D1,n1)+1 = indx(D2,D1,j) or indx(D2,D1,n1)+1 <
indx(D2,D1,j) by XXREAL_0:1;
A315: Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) >=(
lower_bound rng f)*vol(divset(D1,n1+1))
proof
now
per cases by A314,A307;
suppose
A316: indx(D2,D1,j)-indx(D2,D1,n1)=1;
indx(D2,D1,n1)+1 >= 1+1 by A263,XREAL_1:6;
then
A317: indx(D2,D1,n1)+1 <>1;
then upper_bound divset(D2,indx(D2,D1,n1)+1)=D2.(indx
(D2,D1,n1)+1) by A237,A316,INTEGRA1:def 4;
then
A318: upper_bound divset(D2,indx(D2,D1,n1)+1)=D1.j by A10
,A232,A316,INTEGRA1:def 19;
lower_bound divset(D2,indx(D2,D1,n1)+1)=D2.(indx
(D2,D1,n1)+1-1) by A237,A316,A317,INTEGRA1:def 4;
then
A319: lower_bound divset(D2,indx(D2, D1,n1)+1)=D1.n1
by A10,A241,INTEGRA1:def 19;
lower_bound divset(D1,n1+1)=D1.(n1+1-1) by A245,A277
,A266,A279,INTEGRA1:def 4;
then
A320: divset(D2,indx(D2,D1,n1)+1)=divset(D1,n1+1) by A245
,A277,A266,A279,A319,A318,INTEGRA1:def 4;
A321: vol(divset(D2,indx(D2,D1,n1)+1))>=0 by INTEGRA1:9;
reconsider UV = UVf(D2).(indx(D2,D1,n1)+1) as Element of REAL
by XREAL_0:def 1;
1=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A316;
then mid(UVf(D2),indx(D2,D1,n1) +1,indx(D2,D1,j)).1 =
UVf(D2).(1+(indx(D2,D1,n1)+1)-1) by A309,A310,FINSEQ_6:122
.=UVf(D2).(indx(D2,D1,n1)+1);
then
A322: mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))
=<*UV*> by A313,A316,FINSEQ_1:40;
UVf(D2).(indx(D2,D1,n1)+1) =(upper_bound rng(f|
divset(D2,indx(D2,D1,n1)+1))) *vol(divset(D2,indx(D2,D1,n1)+1)) by A237,A316,
INTEGRA1:def 6;
then UVf(D2).(indx(D2,D1,n1)+1) >=(lower_bound rng f)
*vol(divset(D1,n1+1)) by A1,A237,A316,A320,A321,Th19,XREAL_1:64;
hence thesis by A322,FINSOP_1:11;
end;
suppose
A323: indx(D2,D1,j)-indx(D2,D1,n1)=2;
indx(D2,D1,n1)+2 >= 2+1 by A263,XREAL_1:6;
then
A324: indx(D2,D1,n1)+2 <> 1;
then
A325: upper_bound divset(D2,indx(D2,D1,n1)+2)= D2.indx
(D2,D1,j) by A237,A323,INTEGRA1:def 4;
indx(D2,D1,n1)+2-1=indx(D2,D1,n1)+1;
then lower_bound divset(D2,indx(D2,D1,n1)+2) = D2.(
indx(D2,D1,n1)+1) by A237,A323,A324,INTEGRA1:def 4;
then
A326: vol(divset(D2,indx(D2,D1,n1)+2)) =D1.j-D2.(indx(
D2,D1,n1)+1) by A234,A325,INTEGRA1:def 5;
A327: upper_bound divset(D1,n1+1)=D1.(n1+1) by A245,A277,A266
,A279,INTEGRA1:def 4;
lower_bound divset(D1,n1+1)=D1.(n1+1-1) by A245,A277
,A266,A279,INTEGRA1:def 4;
then
A328: vol(divset(D1,n1+1))=D1.(n1+1)-D1.n1 by A327,
INTEGRA1:def 5;
A329: vol(divset(D2,indx(D2,D1,n1)+2)) >= 0 by INTEGRA1:9;
A330: indx(D2,D1,j) <= len UVf(D2) by A239,INTEGRA1:def 6;
A331: vol(divset(D2,indx(D2,D1,n1)+1)) >= 0 by INTEGRA1:9;
A332: 1 <= indx(D2,D1,n1)+1 by NAT_1:12;
A333: indx(D2,D1,n1)+1 <= indx(D2,D1,n1)+2 by XREAL_1:6;
then indx(D2,D1,n1)+1 <= len D2 by A239,A323,XXREAL_0:2
;
then
A334: indx(D2,D1,n1)+1 in dom D2 by A332,FINSEQ_3:25;
then UVf(D2).(indx(D2,D1,n1 )+1) =(upper_bound rng(f|
divset(D2,indx(D2,D1,n1)+1))) *vol(divset(D2,indx(D2,D1,n1)+1)) by
INTEGRA1:def 6;
then
A335: UVf(D2).(indx(D2,D1,n1)+1) >= (lower_bound rng f
)*vol(divset(D2,indx(D2, D1,n1)+1)) by A1,A334,A331,Th19,XREAL_1:64;
indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1=1+1 by A323;
then
A336: mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).2 =
UVf(D2).(2+(indx(D2,D1,n1)+1)-1) by A332,A333,A330,FINSEQ_6:122
.=UVf(D2).(indx(D2,D1,n1)+0+2);
indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1>=1 by A323;
then mid(UVf(D2),indx(D2,D1,n1) +1,indx(D2,D1,j)).1 =
UVf(D2).(1+(indx(D2,D1,n1)+1)-1) by A323,A332,A333,A330,FINSEQ_6:122
.=UVf(D2).(indx(D2,D1,n1)+1);
then mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =<*
UVf(D2).(indx(D2,D1,n1)+1),UVf(D2).(indx(D2,D1,n1)+2)*> by A313,A323,A336,
FINSEQ_1:44;
then
A337: Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))
=UVf(D2).(indx(D2,D1,n1)+1)+UVf(D2).(indx(D2,D1,n1)+2) by RVSUM_1:77;
A338: indx(D2,D1,n1)+1 > 1 by A263,NAT_1:13;
then
A339: upper_bound divset(D2,indx(D2,D1,n1)+1)=D2.(indx
(D2,D1,n1)+1) by A334,INTEGRA1:def 4;
lower_bound divset(D2,indx(D2,D1,n1)+1) = D2.(
indx(D2,D1,n1)+1-1) by A334,A338,INTEGRA1:def 4;
then
A340: vol(divset(D2,indx(D2,D1,n1)+1)) =D2.(indx(D2,D1
,n1)+1)-D1.n1 by A260,A339,INTEGRA1:def 5;
UVf(D2).(indx(D2,D1,n1 )+2) =(upper_bound rng(f|
divset(D2,indx(D2,D1,n1)+2))) *vol(divset(D2,indx(D2,D1,n1)+2)) by A237,A323,
INTEGRA1:def 6;
then UVf(D2).(indx(D2,D1,n1)+2) >= (lower_bound rng f
)*vol(divset(D2,indx(D2, D1,n1)+2)) by A1,A237,A323,A329,Th19,XREAL_1:64;
then Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))
>= (lower_bound rng f)*vol(divset(D2,indx(D2,D1,n1)+1)) + (lower_bound rng f)*
vol(divset(D2,indx(D2,D1,n1)+2)) by A337,A335,XREAL_1:7;
hence thesis by A279,A340,A326,A328;
end;
end;
hence thesis;
end;
A341: n1+1 <= len UVf(D1) by A265,INTEGRA1:def 6;
j-'(n1+1)+1=j-(n1+1)+1 by A279,XREAL_1:233;
then
A342: len mid(UVf(D1),n1+1,j)= 1 by A244,A279,A341,FINSEQ_6:118;
reconsider uv =(upper_bound rng(f|divset(D1,n1
+1)))* vol(divset(D1,n1+1)) as Element of REAL
by XREAL_0:def 1;
(n1+1)+1 <= j+1 by A258,XREAL_1:6;
then 1 <= j+1-(n1+1) by XREAL_1:19;
then mid(UVf(D1),n1+1,j).1 = UVf(D1).(1+(n1+1)-1) by A244
,A279,A341,FINSEQ_6:122
.=(upper_bound rng(f|divset(D1,n1+1)))*vol(divset(D1,n1+
1)) by A266,INTEGRA1:def 6;
then mid(UVf(D1),n1+1,j) =<*uv*> by A342,FINSEQ_1:40;
then
A343: Sum mid(UVf(D1),n1+1,j) =(upper_bound rng(f|divset(D1,
n1+1)))*vol(divset(D1,n1+1)) by FINSOP_1:11;
divset(D1,n1+1) c= A by A266,INTEGRA1:8;
then
A344: upper_bound rng(f|divset(D1,n1+1)) <= upper_bound rng
f by A1,Lm4;
n1+1 in Seg len D1 by A266,FINSEQ_1:def 3;
then n1+1 in Seg len upper_volume(chi(A,A),D1) by
INTEGRA1:def 6;
then
A345: n1+1 in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3;
vol(divset(D1,n1+1))=upper_volume( chi(A,A),D1).(n1+1)
by A266,INTEGRA1:20;
then vol(divset(D1,n1+1)) in rng upper_volume(chi(A,A),D1)
by A345,FUNCT_1:def 3;
then
A346: vol(divset(D1,n1+1))<=delta(D1) by XXREAL_2:def 8;
upper_bound rng f-lower_bound rng f >= 0 by A1,Lm3,XREAL_1:48
;
then
A347: (upper_bound rng f-lower_bound rng f)* vol(divset(D1,
n1+1)) <=(upper_bound rng f-lower_bound rng f)*delta(D1) by A346,XREAL_1:64;
vol(divset(D1,n1+1)) >= 0 by INTEGRA1:9;
then Sum mid(UVf(D1),n1+1,j) <=(upper_bound rng f)*vol(
divset(D1,n1+1)) by A343,A344,XREAL_1:64;
then Sum mid(UVf(D1),n1+1,j)-Sum mid(UVf(D2 ),indx(D2,D1,n1
)+1, indx(D2, D1,j)) <=(upper_bound rng f)*vol(divset(D1,n1+1))-(lower_bound
rng f)* vol(divset(D1,n1+1)) by A315,XREAL_1:13;
hence thesis by A347,XXREAL_0:2;
end;
suppose
A348: n1+1 < j;
A349: j-'(n1+1) = j-(n1+1) by A258,XREAL_1:233;
then
A350: j-'(n1+1)+1=j-n1;
A351: n1 < n1+1 by NAT_1:13;
then
A352: D1.n1 < D1.(n1+1) by A241,A266,SEQM_3:def 1;
then consider
B being non empty closed_interval Subset of REAL, MD1,MD2
being Division of B such that
A353: D1 .n1=lower_bound B and
upper_bound B=MD2.(len MD2) and
A354: upper_bound B=MD1.(len MD1) and
A355: MD1 <= MD2 and
A356: MD1=mid(D1,n1+1,j) and
A357: MD2=mid(D2,indx(D2,D1,n1+1),indx(D2,D1,j)) by A10,A232,A258
,A266,A276,Th15;
A358: len MD1 = j-'(n1+1)+1 by A258,A261,A264,A244,A265,A356,
FINSEQ_6:118;
then
A359: len MD1 = j-(n1+1)+1 by A258,XREAL_1:233;
then
A360: len MD1+(n1+1)-1 =j;
A361: len MD1 in dom MD1 by FINSEQ_5:6;
then
A362: 1 <= len MD1 by FINSEQ_3:25;
A363: lower_bound divset(MD1,len MD1)=lower_bound divset(D1,
j) & upper_bound divset(MD1,len MD1)=upper_bound divset(D1,j)
proof
per cases;
suppose
A364: len MD1=1;
then
A365: upper_bound divset(MD1,len MD1)=MD1.(len MD1) by A361,
INTEGRA1:def 4;
A366: upper_bound divset(D1,j)=D1.j by A232,A245,A277,
INTEGRA1:def 4;
lower_bound divset(D1,j)=D1.(j-1) by A232,A245,A277,
INTEGRA1:def 4;
hence
thesis by A261,A264,A353,A356,A359,A361,A364,A365,A366,
FINSEQ_6:118,INTEGRA1:def 4;
end;
suppose
A367: len MD1<>1;
then len MD1-1 in dom MD1 by A361,INTEGRA1:7;
then
A368: len MD1-1 >= 1 by FINSEQ_3:25;
len MD1 <= len MD1+1 by NAT_1:11;
then
A369: len MD1 -1 <= len MD1 by XREAL_1:20;
upper_bound divset(MD1,len MD1)=MD1.(len MD1) by A361
,A367,INTEGRA1:def 4;
then
A370: upper_bound divset(MD1,len MD1)=D1.j by A258,A264,A244
,A356,A358,A360,A362,FINSEQ_6:122;
A371: len MD1-1+(n1+1)-1=j-1 by A358,A349;
lower_bound divset(MD1,len MD1)=MD1.( len MD1-1)
by A361,A367,INTEGRA1:def 4;
then lower_bound divset(MD1,len MD1) = D1.( j-1) by A258
,A264,A244,A356,A358,A371,A368,A369,FINSEQ_6:122;
hence thesis by A232,A245,A277,A370,INTEGRA1:def 4;
end;
end;
A372: B c= A
proof
let x1 be object;
A373: rng D1 c= A by INTEGRA1:def 2;
D1.n1 in rng D1 by A241,FUNCT_1:def 3;
then
A374: lower_bound A <= D1.n1 by A373,INTEGRA2:1;
assume
A375: x1 in B;
then reconsider x1 as Real;
A376: x1 <= MD1.(len MD1) by A354,A375,INTEGRA2:1;
D1.j in rng D1 by A232,FUNCT_1:def 3;
then
A377: D1.j <= upper_bound A by A373,INTEGRA2:1;
D1.n1 <= x1 by A353,A375,INTEGRA2:1;
then
A378: lower_bound A <= x1 by A374,XXREAL_0:2;
MD1.(len MD1)=D1.(j-n1-1+(n1+1)) by A258,A278,A264,A244
,A356,A358,A349,FINSEQ_6:122
.=D1.j;
then x1 <= upper_bound A by A376,A377,XXREAL_0:2;
hence thesis by A378,INTEGRA2:1;
end;
then reconsider g=f|B as Function of B,REAL by FUNCT_2:32;
A379: delta(MD1) >= 0 by Th9;
A380: g|B is bounded
proof
consider a be Real such that
A381: for x being object st x in A /\ dom f holds a<=f.x
by A1,RFUNCT_1:71;
for x being object st x in B /\ dom g holds a<=g.x
proof
let x be object;
A382: dom f /\ B c= dom f /\ A by A372,XBOOLE_1:26;
assume x in B /\ dom g;
then
A383: x in dom g by XBOOLE_0:def 4;
then x in dom f /\ B by RELAT_1:61;
then a <= f.x by A381,A382;
hence thesis by A383,FUNCT_1:47;
end;
then
A384: g|B is bounded_below by RFUNCT_1:71;
consider a be Real such that
A385: for x being object st x in A /\ dom f holds f.x<=a
by A1,RFUNCT_1:70;
for x being object st x in B /\ dom g holds g.x<=a
proof
let x be object;
A386: dom f /\ B c= dom f /\ A by A372,XBOOLE_1:26;
assume x in B /\ dom g;
then
A387: x in dom g by XBOOLE_0:def 4;
then x in dom f /\ B by RELAT_1:61;
then a >= f.x by A385,A386;
hence thesis by A387,FUNCT_1:47;
end;
then g|B is bounded_above by RFUNCT_1:70;
hence thesis by A384;
end;
lower_bound divset(D1,j) <= D.(i+1) by A233,INTEGRA2:1;
then
A388: D1.(j-1) <= D.(i+1) by A232,A245,A277,INTEGRA1:def 4;
A389: j-'(n1+1)+1=j-(n1+1)+1 by A258,XREAL_1:233;
A390: len upper_volume(g,MD1)=len MD1 by INTEGRA1:def 6
.=j-(n1+1)+1 by A258,A261,A264,A244,A265,A356,A389,
FINSEQ_6:118;
A391: j <= len UVf(D1) by A264,INTEGRA1:def 6;
A392: for k be Nat st 1<=k & k<=len upper_volume(g,MD1)
holds upper_volume(g,MD1).k=mid(UVf(D1),n1+1,j).k
proof
let k be Nat;
assume that
A393: 1 <= k and
A394: k <= len upper_volume(g,MD1);
k in Seg len upper_volume( g,MD1) by A393,A394,FINSEQ_1:1;
then
A395: k in Seg len MD1 by INTEGRA1:def 6;
then
A396: k in dom MD1 by FINSEQ_1:def 3;
k in dom MD1 by A395,FINSEQ_1:def 3;
then
A397: upper_volume(g,MD1).k =(upper_bound rng(g|divset(MD1
,k)))*vol(divset(MD1,k)) by INTEGRA1:def 6;
consider k2 being Element of NAT such that
A398: n1+1=1+k2;
A399: 1 <= k+k2 by A393,NAT_1:12;
k <= j-((n1+1)-1) by A390,A394;
then k+((n1+1)-1) <= j by XREAL_1:19;
then k+k2 <= len D1 by A264,A398,XXREAL_0:2;
then
A400: k+k2 in Seg len D1 by A399,FINSEQ_1:1;
then
A401: k+k2 in dom D1 by FINSEQ_1:def 3;
1+1 <= k+k2 by A259,A393,A398,XREAL_1:7;
then
A402: 1 < k+k2 by NAT_1:13;
A403: k2=(n1+1)-1 by A398;
A404: lower_bound divset(D1,k+k2)=lower_bound divset(MD1,
k) & upper_bound divset(D1,k+k2)= upper_bound divset(MD1,k)
proof
per cases;
suppose
A405: k=1;
then upper_bound divset(MD1,k)=MD1.k by A396,
INTEGRA1:def 4;
then
A406: upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A258,A264
,A244,A356,A390,A393,A394,FINSEQ_6:122;
lower_bound divset(MD1,k)=D1.n1 by A353,A396,A405,
INTEGRA1:def 4;
hence thesis by A403,A402,A401,A405,A406,INTEGRA1:def 4
;
end;
suppose
A407: k<>1;
then upper_bound divset(MD1,k)=MD1.k by A396,
INTEGRA1:def 4;
then
A408: upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A258,A264
,A244,A356,A390,A393,A394,FINSEQ_6:122;
A409: k -1 <= j-(n1+1)+1 by A390,A394,XREAL_1:146,XXREAL_0:2;
A410: lower_bound divset(MD1,k)=MD1.(k-1) by A396,A407,
INTEGRA1:def 4;
A411: k-1 in dom MD1 by A396,A407,INTEGRA1:7;
then 1 <= k-1 by FINSEQ_3:25;
then lower_bound divset(MD1,k)=D1.(k-1+(n1+1)-1) by
A258,A264,A244,A356,A411,A409,A410,FINSEQ_6:122;
hence thesis by A398,A402,A401,A408,INTEGRA1:def 4;
end;
end;
divset(MD1,k)= [.lower_bound divset(MD1,k),
upper_bound divset(MD1,k).] by INTEGRA1:4;
then
A412: divset(D1,k+k2)=divset(MD1,k) by A404,INTEGRA1:4;
A413: k+k2 in dom D1 by A400,FINSEQ_1:def 3;
A414: mid(UVf(D1),n1+1,j).k =UVf(D1).(k+(n1+1)-1) by A258,A244
,A390,A391,A393,A394,FINSEQ_6:122
.=(upper_bound rng(f|divset(D1,k+k2)))* vol(divset(D1,
k+k2)) by A398,A413,INTEGRA1:def 6;
k in dom MD1 by A395,FINSEQ_1:def 3;
then divset(D1,k+k2) c= B by A412,INTEGRA1:8;
hence thesis by A397,A414,A412,FUNCT_1:51;
end;
n1+1 <= len UVf(D1) by A265,INTEGRA1:def 6;
then len upper_volume(g,MD1)=len mid(UVf(D1),n1+1,j) by A258
,A261,A244,A389,A390,A391,FINSEQ_6:118;
then
A415: Sum upper_volume(g,MD1)=Sum mid(UVf(D1),n1+1,j) by A392,
FINSEQ_1:14;
A416: n1 < j-1 by A348,XREAL_1:20;
A417: 1 <= indx(D2,D1,n1)+1 by A263,NAT_1:13;
A418: len MD1 in dom MD1 by FINSEQ_5:6;
A419: upper_bound divset(MD1,len MD1)=MD1.(len MD1)
proof
per cases;
suppose
len MD1 = 1;
hence thesis by A418,INTEGRA1:def 4;
end;
suppose
len MD1 <> 1;
hence thesis by A418,INTEGRA1:def 4;
end;
end;
vol(B)=upper_bound B-D1.n1 by A353,INTEGRA1:def 5;
then vol(B)=D1.j-D1.n1 by A232,A245,A277,A354,A363,A419,
INTEGRA1:def 4;
then
A420: vol(B)<>0 by A232,A241,A245,SEQM_3:def 1;
rng f is bounded_below by A1,INTEGRA1:11;
then
A421: lower_bound rng f <= lower_bound rng g by RELAT_1:70,SEQ_4:47
;
rng f is bounded_above by A1,INTEGRA1:13;
then upper_bound rng f >= upper_bound rng g by RELAT_1:70
,SEQ_4:48;
then upper_bound rng f-lower_bound rng f>= upper_bound rng
g - lower_bound rng g by A421,XREAL_1:13;
then
A422: (upper_bound rng f-lower_bound rng f)*delta(MD1)>= (
upper_bound rng g-lower_bound rng g)*delta(MD1 ) by A379,XREAL_1:64;
D1.n1 < D1.(n1+1) by A241,A266,A351,SEQM_3:def 1;
then indx(D2,D1,n1) < indx(D2,D1,n1+1) by A262,A260,A270,A267
,SEQ_4:137;
then
A423: indx(D2,D1,n1)+1 <= indx(D2,D1,n1+1) by NAT_1:13;
then
A424: indx(D2,D1,n1)+1 <= len D2 by A274,XXREAL_0:2;
A425: indx(D2,D1,n1) < indx(D2,D1,n1)+1 by NAT_1:13;
A426: indx(D2,D1,n1+1)=indx(D2,D1,n1)+1
proof
assume indx(D2,D1,n1+1)<> indx(D2,D1,n1)+1;
then
A427: indx(D2,D1,n1+1)>indx(D2,D1,n1)+1 by A423,XXREAL_0:1;
A428: indx(D2,D1,n1)+1 in dom D2 by A417,A424,FINSEQ_3:25;
then
A429: D2.(indx(D2,D1,n1)+1) in rng D2 by FUNCT_1:def 3;
now
per cases by A11,A429,XBOOLE_0:def 3;
suppose
D2.(indx(D2,D1,n1)+1) in rng D1;
then consider n2 being Element of NAT such that
A430: n2 in dom D1 and
A431: D2.(indx(D2,D1,n1)+1) = D1.n2 by PARTFUN1:3;
D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+1) by A262,A425
,A428,SEQM_3:def 1;
then n1 < n2 by A241,A260,A430,A431,SEQ_4:137;
then
A432: n1+1 <= n2 by NAT_1:13;
D1.n2 < D1.(n1+1) by A270,A267,A427,A428,A431,
SEQM_3:def 1;
hence contradiction by A266,A430,A432,SEQ_4:137;
end;
suppose
A433: D2.(indx(D2,D1,n1)+1) in rng D;
A434: D.i <= upper_bound divset(D1,n1) by A242,INTEGRA2:1;
A435: upper_bound divset(D1,n1)=D1.n1
proof
per cases;
suppose
n1=1;
hence thesis by A241,INTEGRA1:def 4;
end;
suppose
n1<>1;
hence thesis by A241,INTEGRA1:def 4;
end;
end;
consider n2 being Element of NAT such that
A436: n2 in dom D and
A437: D2.(indx(D2,D1,n1)+1) = D.n2 by A433,PARTFUN1:3;
D1.n1 < D.n2 by A262,A260,A425,A428,A437,SEQM_3:def 1;
then D.i < D.n2 by A434,A435,XXREAL_0:2;
then i < n2 by A236,A436,SEQ_4:137;
then
A438: i+1 <= n2 by NAT_1:13;
n1+1 +1 <= j by A348,NAT_1:13;
then
A439: n1+1 <= j-1 by XREAL_1:19;
j -1 in dom D1 by A232,A245,A277,INTEGRA1:7;
then
A440: D1.(n1+1) <= D1.(j-1) by A266,A439,SEQ_4:137;
A441: lower_bound divset(D1,j) <= D.(i+1) by A233,INTEGRA2:1;
lower_bound divset(D1,j) = D1.(j-1) by A232,A245,A277,
INTEGRA1:def 4;
then
A442: D1.(n1+1) <= D.(i+1) by A440,A441,XXREAL_0:2;
D.n2 < D1.(n1+1) by A270,A267,A427,A428,A437,
SEQM_3:def 1;
then D.n2 < D.(i+1) by A442,XXREAL_0:2;
hence contradiction by A231,A436,A438,SEQ_4:137;
end;
end;
hence contradiction;
end;
A443: len MD2=indx(D2,D1,j)-'indx(D2,D1,n1+1)+1 by A275,A271,A274
,A238,A239,A357,FINSEQ_6:118;
then
A444: len MD2 = indx(D2,D1,j)-indx(D2,D1,n1+1)+1 by A275,
XREAL_1:233;
then
A445: len upper_volume(g,MD2)=indx(D2,D1,j)-(indx(D2,D1,n1)+
1)+1 by A426,INTEGRA1:def 6;
for x1 being object holds
x1 in rng MD1 \/ {D.(i+1)} implies x1 in rng MD2
proof let x1 be object;
assume
A446: x1 in rng MD1 \/ {D.(i+1)};
then reconsider x1 as Real;
now
per cases by A446,XBOOLE_0:def 3;
suppose
A447: x1 in rng MD1;
rng MD1 <> {};
then 1 in dom MD1 by FINSEQ_3:32;
then
A448: 1 <= len MD1 by FINSEQ_3:25;
A449: len MD1=j-'(n1+1)+1 by A258,A261,A264,A244,A265,A356,
FINSEQ_6:118;
then len MD1+(n1+1)-1=j-(n1+1)+1+(n1+1)-1 by A258,
XREAL_1:233
.=j;
then
A450: MD1.(len MD1)=D1.j by A258,A264,A244,A356,A448,A449,
FINSEQ_6:122;
rng MD1 c= rng D1 by A356,FINSEQ_6:119;
then
A451: x1 in rng D1 by A447;
rng D1 c= rng D2 by A10,INTEGRA1:def 18;
then consider k such that
A452: k in dom D2 and
A453: D2.k = x1 by A451,PARTFUN1:3;
x1 <= MD1.(len MD1) by A447,Th16;
then k<=indx(D2,D1,j) by A237,A234,A450,A452,A453,
SEQM_3:def 1;
then k-indx(D2,D1,n1+1)<=indx(D2,D1,j)-indx(D2,D1,n1+
1) by XREAL_1:9;
then
A454: k -indx(D2,D1,n1+1)+1<=indx(D2,D1,j)-indx(D2,D1,
n1+1)+1 by XREAL_1:6;
A455: MD1.1 <= x1 by A447,Th16;
MD1.1=D1.(n1+1) by A261,A264,A244,A265,A356,
FINSEQ_6:118;
then
A456: indx(D2,D1,n1+1)<=k by A270,A267,A455,A452,A453,
SEQM_3:def 1;
then consider n being Nat such that
A457: k+1 = indx(D2,D1,n1+1)+n by NAT_1:10,12;
A458: n+indx(D2,D1,n1+1)-1=k by A457;
indx(D2,D1,n1+1)+1<=k+1 by A456,XREAL_1:6;
then
A459: 1 <= k+1-indx(D2,D1,n1 +1) by XREAL_1:19;
then n in dom MD2 by A444,A454,A457,FINSEQ_3:25;
then MD2.n in rng MD2 by FUNCT_1:def 3;
hence x1 in rng MD2 by A275,A271,A239,A357,A453,A459
,A454,A458,FINSEQ_6:122;
end;
suppose
x1 in {D.(i+1)};
then
A460: x1 = D.(i+1) by TARSKI:def 1;
reconsider j1 = j-1 as Element of NAT by A232,A245,A277
,INTEGRA1:7;
A461: rng D c= rng D2 by A9,INTEGRA1:def 18;
D.(i+1) in rng D by A231,FUNCT_1:def 3;
then consider k such that
A462: k in dom D2 and
A463: x1 = D2.k by A460,A461,PARTFUN1:3;
D.(i+1)<= upper_bound divset(D1,j ) by A233,INTEGRA2:1;
then x1 <= D1.j by A232,A245,A277,A460,INTEGRA1:def 4;
then
A464: D2.k <= D2.indx(D2,D1,j) by A10,A232,A463,
INTEGRA1:def 19;
n1<j1 by A348,XREAL_1:20;
then
A465: n1+1 <= j1 by NAT_1:13;
j -1 in dom D1 by A232,A245,A277,INTEGRA1:7;
then
A466: D1.(n1+1) <= D1.(j-1) by A266,A465,SEQ_4:137;
lower_bound divset(D1,j)<=D.(i+1) by A233,INTEGRA2:1;
then D1.(j-1) <= x1 by A232,A245,A277,A460,
INTEGRA1:def 4;
then D2.indx(D2,D1,n1+1) <= D2.k by A267,A463,A466,
XXREAL_0:2;
hence x1 in rng MD2 by A270,A237,A357,A462,A463,A464
,Th17;
end;
end;
hence thesis;
end;
then
A467: rng MD1 \/ {D.(i+1)} c= rng MD2;
rng MD2 <> {};
then 1 in dom MD2 by FINSEQ_3:32;
then
A468: 1 <= len MD2 by FINSEQ_3:25;
A469: len MD2-1+indx(D2,D1,n1+1)=indx(D2,D1, j) by A444;
for x1 being object holds
x1 in rng MD2 implies x1 in (rng MD1 \/ {D.(i+1)})
proof let x1 be object;
assume
A470: x1 in rng MD2;
then reconsider x1 as Real;
A471: MD2.1 <= x1 by A470,Th16;
A472: MD2 .(len MD2)=D2.indx(D2,D1,j) by A275,A271,A239,A357,A468
,A443,A469,FINSEQ_6:122;
A473: rng MD2 c= rng D2 by A357,FINSEQ_6:119;
A474: MD2.1=D2.indx(D2,D1,n1+1) by A271,A274,A238,A239,A357,
FINSEQ_6:118;
A475: x1 <= MD2.(len MD2) by A470,Th16;
then
A476: x1 <= D1.j by A234,A275,A271,A239,A357,A468,A443,A469,
FINSEQ_6:122;
now
per cases by A11,A470,A473,XBOOLE_0:def 3;
suppose
x1 in rng D1;
then consider k such that
A477: k in dom D1 and
A478: D1.k = x1 by PARTFUN1:3;
A479: n1 +1 <= k by A266,A267,A471,A474,A477,A478,
SEQM_3:def 1;
then
A480: 1 <= k-n1 by XREAL_1:19;
n1 <= n1+1 by NAT_1:11;
then consider n being Nat such that
A481: k=n1+n by A479,NAT_1:10,XXREAL_0:2;
A482: k <= j by A232,A234,A475,A472,A477,A478,SEQM_3:def 1;
then
A483: k-n1 <= j- n1 by XREAL_1:9;
A484: 1 <= k-n1 by A479,XREAL_1:19;
A485: j-(n1+1)+1=j-n1;
k-n1 <= len MD1 by A358,A350,A482,XREAL_1:9;
then n in dom MD1 by A484,A481,FINSEQ_3:25;
then
A486: MD1.n in rng MD1 by FUNCT_1:def 3;
MD1.n = D1.(k-n1-1+(n1+1)) by A258,A264,A244,A356
,A480,A483,A485,A481,FINSEQ_6:122
.= D1.k;
hence x1 in (rng MD1 \/ {D.(i+1)}) by A478,A486,
XBOOLE_0:def 3;
end;
suppose
x1 in rng D;
then consider n such that
A487: n in dom D and
A488: D.n = x1 by PARTFUN1:3;
A489: not i+1 < n
proof
j=1 or j <> 1;
then
A490: upper_bound divset(D1,j)=D1.j by A232,INTEGRA1:def 4;
reconsider y1=D.(i+1)as Real;
A491: D.n in rng D by A487,FUNCT_1:def 3;
assume i+1 < n;
then
A492: D.(i+1) < D.n by A231,A487,SEQM_3:def 1;
lower_bound divset(D1,j) <= D.(i+1) by A233,
INTEGRA2:1;
then lower_bound divset(D1,j) <= D.n by A492,
XXREAL_0:2;
then D.n in divset(D1,j) by A476,A488,A490,INTEGRA2:1
;
then
A493: x1 in rng D /\ divset(D1,j) by A488,A491,
XBOOLE_0:def 4;
D.(i+1) in rng D by A231,FUNCT_1:def 3;
then y1 in rng D /\ divset(D1,j) by A233,
XBOOLE_0:def 4;
hence contradiction by A8,A232,A488,A492,A493,Th5;
end;
A494: upper_bound divset(D1,n1)=D1.n1
proof
per cases;
suppose
n1=1;
hence thesis by A241,INTEGRA1:def 4;
end;
suppose
n1<>1;
hence thesis by A241,INTEGRA1:def 4;
end;
end;
D.i <= upper_bound divset(D1,n1) by A242,INTEGRA2:1;
then D.i < D1.(n1+1) by A352,A494,XXREAL_0:2;
then D.i < D.n by A267,A471,A474,A488,XXREAL_0:2;
then i < n by A236,A487,SEQ_4:137;
then i+1 <= n by NAT_1:13;
then i+1 = n or i+1 < n by XXREAL_0:1;
then x1 in {D.(i+1)} by A488,A489,TARSKI:def 1;
hence x1 in (rng MD1 \/ {D.(i+1)}) by XBOOLE_0:def 3;
end;
end;
hence thesis;
end;
then rng MD2 c= rng MD1 \/ {D.(i+1)};
then
A495: rng MD2 = rng MD1 \/ {D.(i+1)} by A467,XBOOLE_0:def 10;
delta(MD1) in rng upper_volume(chi(B,B),MD1) by
XXREAL_2:def 8;
then consider k such that
A496: k in dom upper_volume(chi(B,B),MD1) and
A497: upper_volume(chi(B,B),MD1).k = delta(MD1) by PARTFUN1:3;
A498: k in Seg len upper_volume(chi(B,B),MD1) by A496,
FINSEQ_1:def 3;
then
A499: k in Seg len MD1 by INTEGRA1:def 6;
then
A500: k in dom MD1 by FINSEQ_1:def 3;
A501: k <= len MD1 by A499,FINSEQ_1:1;
then k+n1 <= j by A358,A350,XREAL_1:19;
then
A502: k+n1 <= len D1 by A264,XXREAL_0:2;
A503: 1 <= k by A498,FINSEQ_1:1;
A504: n1+1>1 by A277,NAT_1:13;
then n1>1-1 by XREAL_1:19;
then
A505: k < k+n1 by XREAL_1:29;
then 1 < k+n1 by A503,XXREAL_0:2;
then
A506: k+n1 in dom D1 by A502,FINSEQ_3:25;
lower_bound divset(MD1,k)=lower_bound divset(D1,k+n1)
& upper_bound divset(MD1,k)=upper_bound divset(D1,k+n1)
proof
per cases;
suppose
A507: k=1;
then upper_bound divset(MD1,k)=MD1.k by A500,
INTEGRA1:def 4;
then
A508: upper_bound divset(MD1,k)= D1.(k+(n1+1)-1) by A258,A264
,A244,A356,A358,A503,A501,FINSEQ_6:122;
lower_bound divset(D1,k+n1)=D1.(k+n1-1) by A503,A505,A506
,INTEGRA1:def 4;
hence thesis by A353,A504,A500,A506,A507,A508,
INTEGRA1:def 4;
end;
suppose
A509: k<>1;
then upper_bound divset(MD1,k)=MD1.k by A500,
INTEGRA1:def 4;
then
A510: upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A258,A264
,A244,A356,A358,A503,A501,FINSEQ_6:122;
A511: lower_bound divset(MD1,k)=MD1.(k-1) by A500,A509,
INTEGRA1:def 4;
A512: k-1 in dom MD1 by A500,A509,INTEGRA1:7;
then
A513: k-1 <= len MD1 by FINSEQ_3:25;
1 <= k-1 by A512,FINSEQ_3:25;
then lower_bound divset(MD1,k)=D1.(k-1+(n1+1)-1) by A258
,A264,A244,A356,A358,A512,A513,A511,FINSEQ_6:122;
hence thesis by A503,A505,A506,A510,INTEGRA1:def 4;
end;
end;
then divset(MD1,k)=[.lower_bound divset(D1,k+n1),
upper_bound divset(D1,k+n1).] by INTEGRA1:4;
then
A514: divset(MD1,k)=divset(D1,k+n1) by INTEGRA1:4;
k+n1 in Seg len D1 by A506,FINSEQ_1:def 3;
then k+n1 in Seg len upper_volume(chi(A,A),D1) by
INTEGRA1:def 6;
then
A515: k+n1 in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3;
k in dom MD1 by A499,FINSEQ_1:def 3;
then delta(MD1) = vol(divset(MD1,k)) by A497,INTEGRA1:20;
then delta(MD1)=upper_volume(chi(A,A),D1).( k+n1) by A506
,A514,INTEGRA1:20;
then delta(MD1) in rng upper_volume(chi(A,A),D1) by A515,
FUNCT_1:def 3;
then delta(MD1) <= max rng upper_volume(chi(A,A),D1) by
XXREAL_2:def 8;
then
A516: delta(MD1) <= delta(D1);
A517: D.(i+1)<= upper_bound divset(D1,j) by A233,INTEGRA2:1;
lower_bound divset(D1,j)<=D.(i+1) by A233,INTEGRA2:1;
then
A518: D.(i+1) in divset(MD1,len MD1) by A363,A517,INTEGRA2:1;
j-1 in dom D1 by A232,A245,A277,INTEGRA1:7;
then D1.n1 <D1.(j-1) by A241,A416,SEQM_3:def 1;
then D.(i+1) > lower_bound B by A353,A388,XXREAL_0:2;
then Sum upper_volume(g,MD1)-Sum upper_volume(g,MD2)<= (
upper_bound rng g- lower_bound rng g)*delta(MD1) by A355,A380,A495,A518,A420
,Th14;
then
A519: Sum upper_volume(g,MD1)-Sum upper_volume(g,MD2)<= (
upper_bound rng f-lower_bound rng f)*delta(MD1) by A422,XXREAL_0:2;
A520: indx(D2,D1,j) <= len UVf(D2) by A239,INTEGRA1:def 6;
A521: indx (D2,D1,n1)+1 <= indx(D2,D1,j) by A275,A423,XXREAL_0:2;
A522: for k be Nat st 1<=k & k<=len upper_volume(g,MD2)
holds upper_volume(g,MD2).k=mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k
proof
let k be Nat;
assume that
A523: 1 <= k and
A524: k <= len upper_volume(g,MD2);
A525: k in Seg len upper_volume( g,MD2) by A523,A524,FINSEQ_1:1;
A526: mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k =UVf(
D2).(k+(indx(D2,D1,n1)+1)-1) by A417,A445,A520,A521,A523,A524,FINSEQ_6:122;
A527: k in Seg len MD2 by A525,INTEGRA1:def 6;
then k in dom MD2 by FINSEQ_1:def 3;
then
A528: upper_volume(g,MD2).k =(upper_bound rng(g|divset(MD2
,k)))*vol(divset(MD2,k)) by INTEGRA1:def 6;
1<=indx(D2,D1,n1)+1 by NAT_1:12;
then 1+1<=k+(indx(D2,D1,n1)+1) by A523,XREAL_1:7;
then
A529: 1 <= k+(indx(D2,D1,n1)+1)- 1 by XREAL_1:19;
consider k2 being Element of NAT such that
A530: indx(D2,D1,n1)+1=1+k2;
k <= indx(D2,D1,j)-((indx(D2,D1,n1)+1)-1) by A444,A426,A524
,INTEGRA1:def 6;
then k+((indx(D2,D1,n1)+1)-1) <= indx(D2,D1,j) by
XREAL_1:19;
then k +(indx(D2,D1,n1)+1)-1 <= len UVf(D2) by A520,
XXREAL_0:2;
then k+k2 in Seg len UVf(D2) by A529,A530,FINSEQ_1:1;
then
A531: k+k2 in Seg len D2 by INTEGRA1:def 6;
then k+k2 in dom D2 by FINSEQ_1:def 3;
then
A532: mid(UVf(D2),indx(D2,D1,n1) +1,indx(D2,D1,j)).k =(
upper_bound rng(f|divset(D2,k+k2)))* vol(divset(D2,k+k2)) by A526,A530,
INTEGRA1:def 6;
A533: lower_bound divset(MD2,k)=lower_bound divset(D2,k+k2
) & upper_bound divset(MD2,k)=upper_bound divset(D2,k+k2)
proof
k+k2>=1+1 by A263,A523,A530,XREAL_1:7;
then
A534: k+k2>1 by NAT_1:13;
A535: k in dom MD2 by A527,FINSEQ_1:def 3;
A536: k+k2 in dom D2 by A531,FINSEQ_1:def 3;
per cases;
suppose
A537: k=1;
then
A538: upper_bound divset(D2,k+k2)=D2.(1+ k2 ) by A534,A536,
INTEGRA1:def 4;
A539: lower_bound divset(MD2,k)= lower_bound B by A535,A537,
INTEGRA1:def 4;
upper_bound divset(MD2,k)=MD2.k by A535,A537,
INTEGRA1:def 4;
then
A540: upper_bound divset(MD2,k)= D2.(1+indx(D2,D1,n1+1
)-1) by A275,A239,A357,A417,A426,A445,A524,A537,FINSEQ_6:122
.=D1.(n1+1) by A10,A266,INTEGRA1:def 19;
lower_bound divset(D2,k+k2)=D2.(1+k2-1) by A534,A536
,A537,INTEGRA1:def 4;
hence
thesis by A10,A241,A266,A353,A426,A530,A539,A540,A538,
INTEGRA1:def 19;
end;
suppose
A541: k<>1;
then upper_bound divset(MD2,k)=MD2.k by A535,
INTEGRA1:def 4;
then
A542: upper_bound divset(MD2,k)= D2.(k+(indx(D2,D1,n1)
+1)-1) by A275,A239,A357,A417,A426,A445,A523,A524,FINSEQ_6:122;
A543: k -1 <= indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A445,A524
,XREAL_1:146,XXREAL_0:2;
A544: lower_bound divset(MD2,k)=MD2.(k-1) by A535,A541,
INTEGRA1:def 4;
A545: k-1 in dom MD2 by A535,A541,INTEGRA1:7;
then 1 <= k-1 by FINSEQ_3:25;
then lower_bound divset(MD2,k)=D2.(k-1+(indx(D2,D1,n1
)+1)-1) by A275,A239,A357,A417,A426,A545,A543,A544,FINSEQ_6:122;
hence thesis by A530,A534,A536,A542,INTEGRA1:def 4;
end;
end;
divset (MD2,k)=[.lower_bound divset(MD2,k),
upper_bound divset(MD2,k).] by INTEGRA1:4;
then
A546: divset(MD2,k)=divset(D2,k+ k2) by A533,INTEGRA1:4;
k in dom MD2 by A527,FINSEQ_1:def 3;
then divset(D2,k+k2) c= B by A546,INTEGRA1:8;
hence thesis by A528,A532,A546,FUNCT_1:51;
end;
indx(D2,D1,n1)+1 <= len UVf(D2) by A424,INTEGRA1:def 6;
then len mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) = indx
(D2,D1,j)-'(indx(D2,D1,n1)+1)+1 by A238,A417,A520,A521,FINSEQ_6:118;
then len upper_volume(g,MD2) =len mid(UVf(D2),indx(D2,D1,n1
)+1,indx(D2,D1,j)) by A275,A423,A445,XREAL_1:233,XXREAL_0:2;
then
A547: Sum upper_volume(g,MD2)=Sum mid(UVf(D2),indx(D2,D1,n1)
+1,indx(D2,D1,j)) by A522,FINSEQ_1:14;
upper_bound rng f - lower_bound rng f >= 0 by A1,Lm3,
XREAL_1:48;
then (upper_bound rng f-lower_bound rng f)*delta(MD1) <= (
upper_bound rng f-lower_bound rng f)* delta(D1) by A516,XREAL_1:64;
hence thesis by A519,A547,A415,XXREAL_0:2;
end;
end;
then
A548: (PUf(D1,n1)-PUf(D2,indx(D2,D1,n1)))+ (Sum mid(UVf(D1),n1+
1,j)-Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))) <=i*(upper_bound rng f-
lower_bound rng f)*delta(D1)+(upper_bound rng f- lower_bound rng f)*delta(D1)
by A243,XREAL_1:7;
n1 < n1+1 by NAT_1:13;
then D1.n1 < D1.(n1+1) by A241,A266,SEQM_3:def 1;
then indx(D2,D1,n1) < indx(D2,D1,n1+1) by A262,A260,A270,A267,
SEQ_4:137;
then
A549: indx(D2,D1,n1)<indx(D2,D1,j) by A275,XXREAL_0:2;
indx(D2,D1,n1) in Seg len D2 by A262,FINSEQ_1:def 3;
then indx(D2,D1,n1) in Seg len UVf(D2) by INTEGRA1:def 6;
then indx(D2,D1,n1) in dom UVf(D2) by FINSEQ_1:def 3;
then PUf(D2,indx(D2,D1,n1))=Sum(UVf(D2)|indx(D2,D1,n1)) by
INTEGRA1:def 20
.=Sum mid(UVf(D2),1,indx(D2,D1,n1)) by A263,FINSEQ_6:116;
then PUf(D2,indx(D2,D1,n1))+Sum mid(UVf(D2),indx(D2,D1,n1)+1,
indx(D2, D1,j)) =Sum(mid(UVf(D2),1,indx(D2,D1,n1))^ mid(UVf(D2),indx(D2,D1,n1)+
1,indx(D2,D1,j))) by RVSUM_1:75
.=Sum mid(UVf(D2),1,indx(D2,D1,j)) by A263,A549,A240,INTEGRA2:4
.=Sum(UVf(D2)|indx(D2,D1,j)) by A238,FINSEQ_6:116;
then PUf(D2,indx(D2,D1,j))= PUf(D2,indx(D2,D1,n1))+Sum mid(UVf
(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) by A273,INTEGRA1:def 20;
then (PUf(D1,n1)-PUf(D2,indx(D2,D1,n1)))+ (Sum mid(UVf(D1),n1+
1,j)-Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))) =PUf(D1,j)-PUf(D2,indx(D2
,D1,j)) by A272;
hence thesis by A232,A233,A548;
end;
hence thesis;
end;
for k being non zero Nat holds P[k] from NAT_1:sch 10(A36,
A227);
then P[i];
hence thesis by A14;
end;
A550: len D1 in dom D1 by FINSEQ_5:6;
then D1.(len D1) = D2.indx(D2,D1,len D1) by A10,INTEGRA1:def 19;
then upper_bound A = D2.indx(D2,D1,len D1) by INTEGRA1:def 2;
then
A551: D2.(len D2) = D2.indx(D2,D1,len D1) by INTEGRA1:def 2;
len D in dom D by FINSEQ_5:6;
then consider j such that
A552: j in dom D1 and
A553: D.(len D) in divset(D1,j) and
A554: PUf(D1,j)-PUf(D2,indx(D2,D1,j)) <= (len D)*(upper_bound(rng
f)- lower_bound(rng f))*delta(D1) by A13;
A555: j = len D1
proof
j in Seg len D1 by A552,FINSEQ_1:def 3;
then
A556: j <= len D1 by FINSEQ_1:1;
assume j<>len D1;
then j < len D1 by A556,XXREAL_0:1;
then D1.j < D1.(len D1) by A552,A550,SEQM_3:def 1;
then
A557: D1.j < upper_bound A by INTEGRA1:def 2;
A558: upper_bound divset(D1,j) < upper_bound A
proof
per cases;
suppose
j=1;
hence thesis by A552,A557,INTEGRA1:def 4;
end;
suppose
j<>1;
hence thesis by A552,A557,INTEGRA1:def 4;
end;
end;
D.(len D) <= upper_bound divset(D1,j) by A553,INTEGRA2:1;
hence contradiction by A558,INTEGRA1:def 2;
end;
indx(D2,D1,len D1) in dom D2 by A10,A550,INTEGRA1:def 19;
then indx(D2,D1,len D1)=len D2 by A12,A551,SEQ_4:138;
then upper_sum(f,D1)-PUf(D2,len D2)<= (len D)*(upper_bound(rng f)-
lower_bound(rng f))* delta(D1) by A554,A555,INTEGRA1:42;
hence thesis by INTEGRA1:42;
end;
hence thesis by A9,A10,A11;
end;
hence thesis;
end;
::$N Darboux's Theorem
theorem
f|A is bounded & delta(T) is 0-convergent non-zero & vol(A)<>0 implies
lower_sum(f,T) is convergent & lim lower_sum(f,T) = lower_integral(f)
proof
assume that
A1: f|A is bounded and
A2: delta(T) is 0-convergent non-zero and
A3: vol(A)<>0;
A4: delta(T) is convergent by A2,FDIFF_1:def 1;
A5: for D,D1 ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & 0<=
lower_sum(f,D2)-lower_sum(f,D) & 0<=lower_sum(f,D2)-lower_sum(f,D1)
by A1,Th20;
A10: for D,D1 st delta(D1)<min rng upper_volume(chi(A,A),D) ex D2 st D<=D2 &
D1<=D2 & rng D2=rng D1 \/ rng D & lower_sum(f,D2)-lower_sum(f,D1) <= (len D)*(
upper_bound(rng f)-lower_bound(rng f))*delta(D1) by A1,Th21;
A552: lim delta(T) = 0 by A2,FDIFF_1:def 1;
A553: delta(T) is non-zero by A2;
A554: for e st e>0
ex n being Nat st
for m being Nat st n<=m holds 0 < (delta(T)).m & (delta(T)).m < e
proof
let e;
assume e>0;
then consider n being Nat such that
A555: for m being Nat st n<=m holds |.(delta(T)).m-0.|<e
by A4,A552,SEQ_2:def 7;
take n;
let m be Nat;
reconsider mm=m as Element of NAT by ORDINAL1:def 12;
A556: (delta(T)).m = delta(T.mm) by Def2;
delta(T.mm) in rng upper_volume(chi(A,A),T.mm) by XXREAL_2:def 8;
then consider i such that
A557: i in dom upper_volume(chi(A,A),T.mm) and
A558: delta(T.mm)=upper_volume(chi(A,A),T.mm).i by PARTFUN1:3;
consider D being Division of A such that
A559: D = T.mm;
i in Seg len upper_volume(chi(A,A),T.mm) by A557,FINSEQ_1:def 3;
then i in Seg len D by A559,INTEGRA1:def 6;
then i in dom D by FINSEQ_1:def 3;
then
A560: delta(T.mm)=vol(divset(T.mm,i)) by A558,A559,INTEGRA1:20;
assume n<=m;
then |.(delta(T)).m-0.|<e by A555;
then
A561: (delta(T)).m+|.(delta(T)).m-0.|<e+|.(delta(T)).m-0.| by ABSVALUE:4
,XREAL_1:8;
(delta(T)).m<>0 by A553,SEQ_1:5;
hence thesis by A561,A556,A560,INTEGRA1:9,XREAL_1:6;
end;
A562: for e be Real st e>0
ex n being Nat st for m being Nat st n<=m holds |.(
lower_sum(f,T)).m-lower_integral(f).|<e
proof
set h=lower_bound rng f;
set H=upper_bound rng f;
let e be Real;
assume
A563: e>0;
then
A564: e/2>0 by XREAL_1:139;
reconsider e as Real;
A565: H-h >= 0 by A1,Lm3,XREAL_1:48;
A566: rng lower_sum_set(f) is bounded_above by A1,INTEGRA2:36;
lower_integral(f) = upper_bound rng lower_sum_set(f) by INTEGRA1:def 15;
then consider y be Real such that
A567: y in rng lower_sum_set(f) and
A568: lower_integral(f)-e/2<y by A564,A566,SEQ_4:def 1;
consider D being Division of A such that
D in dom lower_sum_set(f) and
A569: y=(lower_sum_set(f)).D and
A570: D.1>lower_bound A by A3,A567,Lm7;
deffunc F(Nat)=In(vol(divset(D,$1)),REAL);
set p=len D;
consider v being FinSequence of REAL such that
A571: len v = len D & for j be Nat st j in dom v holds v.j=F(j) from
FINSEQ_2:sch 1;
consider v1 being non-decreasing FinSequence of REAL such that
A572: v,v1 are_fiberwise_equipotent by INTEGRA2:3;
defpred P[Nat] means $1 in dom v1 & v1.$1>0;
A573: dom v = Seg len D by A571,FINSEQ_1:def 3;
A574: ex k be Nat st P[k]
proof
consider H being Function such that
dom H = dom v and
rng H = dom v1 and
H is one-to-one and
A575: v=v1*H by A572,CLASSES1:77;
consider k such that
A576: k in dom D and
A577: vol(divset(D,k)) > 0 by A3,Th2;
A578: dom D = Seg len v by A571,FINSEQ_1:def 3;
then H.k in dom v1 by A571,A573,A575,A576,FUNCT_1:11;
then reconsider Hk = H.k as Nat;
v.k = F(k) by A571,A578,A573,A576;
then v.k > 0 by A577;
then P[Hk] by A571,A573,A575,A576,A578,FUNCT_1:11,12;
hence thesis;
end;
consider k be Nat such that
A579: P[k] & for n be Nat st P[n] holds k<=n from NAT_1:sch 5(A574);
A580: 2*p >0 by XREAL_1:129;
then
A581: 2*p*(H-h+1) > 0 by A565,XREAL_1:129;
min(v1.k,e/(2*p*(H-h+1))) > 0
proof
per cases by XXREAL_0:15;
suppose
min(v1.k,e/(2*p*(H-h+1))) = v1.k;
hence thesis by A579;
end;
suppose
min(v1.k,e/(2*p*(H-h+1))) = e/(2*p*(H-h+1));
hence thesis by A563,A581,XREAL_1:139;
end;
end;
then consider n being Nat such that
A582: for m being Nat st n<=m
holds 0 < (delta(T)).m & (delta(T)).m < min(v1.k,
e/(2*p*(H-h+1))) by A554;
take n;
A583: y=lower_sum(f,D) by A569,INTEGRA1:def 11;
A584: v1.1 > 0
proof
A585: for n1 be Element of NAT st n1 in dom D holds vol(divset(D,n1))
>0
proof
let n1 be Element of NAT;
assume
A586: n1 in dom D;
then
A587: 1 <= n1 by FINSEQ_3:25;
per cases by A587,XXREAL_0:1;
suppose
A588: n1=1;
then
A589: upper_bound divset(D,n1)=D.n1 by A586,INTEGRA1:def 4;
lower_bound divset(D,n1)=lower_bound A by A586,A588,INTEGRA1:def 4;
then vol(divset(D,n1))=D.n1-lower_bound A by A589,INTEGRA1:def 5;
hence thesis by A570,A588,XREAL_1:50;
end;
suppose
A590: n1>1;
then
A591: upper_bound divset(D,n1)=D.n1 by A586,INTEGRA1:def 4;
lower_bound divset(D,n1)=D.(n1-1) by A586,A590,INTEGRA1:def 4;
then
A592: vol(divset(D,n1))=D.n1-D.(n1-1) by A591,INTEGRA1:def 5;
n1 < n1+1 by XREAL_1:29;
then
A593: n1-1 < n1 by XREAL_1:19;
n1-1 in dom D by A586,A590,INTEGRA1:7;
then D.(n1-1)<D.n1 by A586,A593,SEQM_3:def 1;
hence thesis by A592,XREAL_1:50;
end;
end;
A594: k <= len v1 by A579,FINSEQ_3:25;
1 <= k by A579,FINSEQ_3:25;
then 1 <= len v1 by A594,XXREAL_0:2;
then
1 in dom v1 by FINSEQ_3:25;
then
A595: v1.1 in rng v1 by FUNCT_1:def 3;
rng v = rng v1 by A572,CLASSES1:75;
then consider n1 being Element of NAT such that
A596: n1 in dom v and
A597: v1.1 = v.n1 by A595,PARTFUN1:3;
n1 in Seg len D by A571,A596,FINSEQ_1:def 3;
then
A598: n1 in dom D by FINSEQ_1:def 3;
v1.1 = F(n1)by A571,A596,A597
.= vol(divset(D,n1));
hence thesis by A585,A598;
end;
A599: v1.k = min rng upper_volume(chi(A,A),D)
proof
A600: k=1
proof
len v1 = len v by A572,RFINSEQ:3;
then k in Seg len v by A579,FINSEQ_1:def 3;
then
A601: 1 <= k by FINSEQ_1:1;
k in Seg len v1 by A579,FINSEQ_1:def 3;
then k <= len v1 by FINSEQ_1:1;
then 1 <= len v1 by A601,XXREAL_0:2;
then
A602: 1 in dom v1 by FINSEQ_3:25;
assume k <> 1;
then k > 1 by A601,XXREAL_0:1;
hence contradiction by A579,A584,A602;
end;
min rng upper_volume(chi(A,A),D) in rng upper_volume(chi(A,A),D
) by XXREAL_2:def 7;
then consider m such that
A603: m in dom upper_volume(chi(A,A),D) and
A604: min rng upper_volume(chi(A,A),D)=upper_volume(chi(A,A),D).m
by PARTFUN1:3;
m in Seg len upper_volume(chi(A,A),D) by A603,FINSEQ_1:def 3;
then
A605: m in Seg len D by INTEGRA1:def 6;
then m in dom D by FINSEQ_1:def 3;
then
A606: min rng upper_volume(chi(A,A),D)=vol(divset(D,m)) by A604,
INTEGRA1:20;
A607: v.m = F(m) by A571,A573,A605
.=min rng upper_volume(chi(A,A),D) by A606;
A608: rng v = rng v1 by A572,CLASSES1:75;
m in dom v by A571,A605,FINSEQ_1:def 3;
then min rng upper_volume(chi(A,A),D) in rng v by A607,FUNCT_1:def 3;
then consider m1 being Element of NAT such that
A609: m1 in dom v1 and
A610: min rng upper_volume(chi(A,A),D)=v1.m1 by A608,PARTFUN1:3;
v1.k in rng v1 by A579,FUNCT_1:def 3;
then consider k2 being Element of NAT such that
A611: k2 in dom v and
A612: v1.k = v.k2 by A608,PARTFUN1:3;
A613: k2 in Seg len D by A571,A611,FINSEQ_1:def 3;
then
A614: k2 in dom D by FINSEQ_1:def 3;
k2 in Seg len upper_volume(chi(A,A),D) by A613,INTEGRA1:def 6;
then
A615: k2 in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3;
v1.k = F(k2) by A571,A611,A612
.= vol(divset(D,k2));
then v1.k = upper_volume(chi(A,A),D).k2 by A614,INTEGRA1:20;
then v1.k in rng upper_volume(chi(A,A),D) by A615,FUNCT_1:def 3;
then
A616: v1.k >= min rng upper_volume(chi(A,A),D) by XXREAL_2:def 7;
m1 >= 1 by A609,FINSEQ_3:25;
then v1.1 <= min rng upper_volume(chi(A,A),D) by A579,A600,A609,A610,
INTEGRA2:2;
hence thesis by A600,A616,XXREAL_0:1;
end;
H-h <= H-h+1 by XREAL_1:29;
then
A617: p*(H-h)<=p*(H-h+1) by XREAL_1:64;
set sD=lower_sum(f,D);
set s=lower_integral(f);
let m be Nat;
reconsider mm=m as Element of NAT by ORDINAL1:def 12;
reconsider D1 = T.mm as Division of A;
A618: min(v1.k,e/(2*p*(H-h+1))) <= e/(2*p*(H-h+1)) by XXREAL_0:17;
assume
A619: n<=m;
then (delta(T)).m < min(v1.k,e/(2*p*(H-h+1))) by A582;
then
A620: delta(D1)<min (v1.k,e/(2*p*(H-h+1))) by Def2;
(delta(T)).m < min(v1.k,e/(2*p*(H-h+1))) by A582,A619;
then (delta(T)).m < e/(2*p*(H-h+1)) by A618,XXREAL_0:2;
then (delta(T)).m*(2*p*(H-h+1))<e by A580,A565,XREAL_1:79,129;
then ((delta(T)).m*(p*(H-h+1)))*2<e;
then
A621: p*(H-h+1)*(delta(T)).m < e/2 by XREAL_1:81;
T.mm in divs A by INTEGRA1:def 3;
then
A622: T.mm in dom lower_sum_set(f) by FUNCT_2:def 1;
(lower_sum(f,T)).mm = lower_sum(f,T.mm) by INTEGRA2:def 3;
then (lower_sum(f,T)).m = (lower_sum_set(f)).(T.m) by INTEGRA1:def 11;
then (lower_sum(f,T)).m in rng lower_sum_set(f) by A622,FUNCT_1:def 3;
then upper_bound rng lower_sum_set(f)>=(lower_sum(f,T)).m by A566,
SEQ_4:def 1;
then lower_integral(f)>=(lower_sum(f,T)).m by INTEGRA1:def 15;
then
A623: lower_integral(f)-(lower_sum(f,T)).m >= 0 by XREAL_1:48;
0 < (delta(T)).m by A582,A619;
then
A624: p*(H-h)*(delta(T)).m <= p*(H-h+1)*(delta(T)).m by A617,XREAL_1:64;
set sD1=lower_sum(f,T.mm);
consider D2 being Division of A such that
A625: D <= D2 and
D1 <= D2 and
A626: rng D2 = rng D1 \/ rng D and
0 <= lower_sum(f,D2)-lower_sum(f,D) and
0 <= lower_sum(f,D2)-lower_sum(f,D1) by A5;
set sD2=lower_sum(f,D2);
A627: (sD-sD1) - (sD2-sD1) = (sD-sD2);
min (v1.k,e/(2*p*(H-h+1))) <= v1.k by XXREAL_0:17;
then delta(D1)<v1.k by A620,XXREAL_0:2;
then ex D3 being Division of A st D<=D3 & D1<=D3 & rng D3= rng D1
\/ rng D & lower_sum(f,D3)-lower_sum(f,D1) <=(len D)*(upper_bound rng f -
lower_bound rng f)*delta(D1) by A10,A599;
then
A628: lower_sum(f,D2)-lower_sum(f,D1)<=(len D)*(upper_bound rng f-
lower_bound rng f)*delta(D1) by A626,Th6;
lower_sum(f,D)-lower_sum(f,D2)<=0 by A1,A625,INTEGRA1:46,XREAL_1:47;
then
A629: sD-sD1 <= sD2-sD1 by A627,XREAL_1:50;
delta(D1)=(delta(T)).m by Def2;
then lower_sum(f,D2)-lower_sum(f,T.mm)
<= p*(H-h+1)*(delta(T)).m by A628
,A624,XXREAL_0:2;
then sD-sD1 <= p*(H-h+1)*(delta(T)).m by A629,XXREAL_0:2;
then sD-sD1 < e/2 by A621,XXREAL_0:2;
then
A630: sD-sD1+e/2 < e/2+e/2 by XREAL_1:6;
s-sD1+sD1 < sD+e/2 by A568,A583,XREAL_1:19;
then s-sD1 < sD+e/2-sD1 by XREAL_1:20;
then s-sD1 < e by A630,XXREAL_0:2;
then lower_integral(f)-(lower_sum(f,T)).m < e by INTEGRA2:def 3;
then |.lower_integral(f)-(lower_sum(f,T)).m.| < e by A623,ABSVALUE:def 1;
then |.-(lower_integral(f)-(lower_sum(f,T)).m).| < e by COMPLEX1:52;
hence thesis;
end;
hence lower_sum(f,T) is convergent by SEQ_2:def 6;
hence thesis by A562,SEQ_2:def 7;
end;
theorem
f|A is bounded & delta(T) is 0-convergent non-zero & vol(A)<>0 implies
upper_sum(f,T) is convergent & lim upper_sum(f,T) = upper_integral(f)
proof
assume
A1: f|A is bounded;
then
A2: for D,D1 ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & 0<=
upper_sum(f,D)-upper_sum(f,D2) & 0<=upper_sum(f,D1)-upper_sum(f,D2) by Th22;
A7: for D,D1 st delta(D1)<min rng upper_volume(chi(A,A),D) holds ex D2 st D
<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & upper_sum(f,D1)-upper_sum(f,D2) <= (
len D)*(upper_bound(rng f)-lower_bound(rng f))*delta(D1) by A1,Th23;
assume
A559: delta(T) is 0-convergent non-zero;
then
A560: delta(T) is convergent by FDIFF_1:def 1;
A561: lim delta(T) = 0 by A559,FDIFF_1:def 1;
assume
A562: vol(A)<>0;
A563: delta(T) is non-zero by A559;
A564: for e st e>0
ex n being Nat st
for m being Nat st n<=m holds 0 < (delta(T)).m & (delta(T)).m
< e
proof
let e;
assume e>0;
then consider n being Nat such that
A565: for m being Nat st n<=m
holds |.(delta(T)).m-0.|<e by A560,A561,SEQ_2:def 7;
take n;
let m be Nat;
reconsider mm =m as Element of NAT by ORDINAL1:def 12;
assume n<=m;
then |.(delta(T)).m-0.|<e by A565;
then
A566: (delta(T)).m+|.(delta(T)).m-0.|<e+|.(delta(T)).m-0.| by ABSVALUE:4
,XREAL_1:8;
reconsider D = T.mm as Division of A;
A567: (delta(T)).m = delta(T.mm) by Def2;
delta(T.mm) in rng upper_volume(chi(A,A),T.mm) by XXREAL_2:def 8;
then consider i such that
A568: i in dom upper_volume(chi(A,A),T.mm) and
A569: delta(T.mm)=upper_volume(chi(A,A),T.mm).i by PARTFUN1:3;
i in Seg len upper_volume(chi(A,A),T.mm) by A568,FINSEQ_1:def 3;
then i in Seg len D by INTEGRA1:def 6;
then i in dom D by FINSEQ_1:def 3;
then
A570: delta(T.mm)=vol(divset(T.mm,i)) by A569,INTEGRA1:20;
(delta(T)).m<>0 by A563,SEQ_1:5;
hence thesis by A566,A567,A570,INTEGRA1:9,XREAL_1:6;
end;
A571: for e be Real st e>0
ex n being Nat st for m being Nat st n<=m holds |.(
upper_sum(f,T)).m-upper_integral(f).|<e
proof
let e be Real;
assume
A572: e>0;
then
A573: e/2>0 by XREAL_1:139;
reconsider e as Real;
A574: rng upper_sum_set(f) is bounded_below by A1,INTEGRA2:35;
upper_integral(f) = lower_bound rng upper_sum_set(f) by INTEGRA1:def 14;
then consider y be Real such that
A575: y in rng upper_sum_set(f) and
A576: upper_integral(f)+e/2>y by A573,A574,SEQ_4:def 2;
ex D being Division of A st D in dom upper_sum_set(f) & y=(
upper_sum_set(f)).D & D.1 > lower_bound A
proof
consider D3 being Element of divs A such that
A577: D3 in dom upper_sum_set(f) and
A578: y=(upper_sum_set(f)).D3 by A575,PARTFUN1:3;
reconsider D3 as Division of A by INTEGRA1:def 3;
A579: len D3 in Seg len D3 by FINSEQ_1:3;
then 1 <= len D3 by FINSEQ_1:1;
then 1 in Seg len D3 by FINSEQ_1:1;
then
A580: 1 in dom D3 by FINSEQ_1:def 3;
per cases;
suppose
A581: D3.1 <> lower_bound A;
D3.1 in A by A580,INTEGRA1:6;
then lower_bound A <= D3.1 by INTEGRA2:1;
then D3.1 > lower_bound A by A581,XXREAL_0:1;
hence thesis by A577,A578;
end;
suppose
A582: D3.1 = lower_bound A;
ex D being Division of A st D in dom upper_sum_set(f) & y=(
upper_sum_set(f)).D & D.1 > lower_bound A
proof
A583: upper_volume(f,D3).1= (upper_bound rng(f|divset(D3,1)))*vol(
divset(D3,1)) by A580,INTEGRA1:def 6;
vol(A) >= 0 by INTEGRA1:9;
then
A584: upper_bound A - lower_bound A > 0 by A562,INTEGRA1:def 5;
A585: y=upper_sum(f,D3) by A578,INTEGRA1:def 10
.=Sum(upper_volume(f,D3)) by INTEGRA1:def 8
.=Sum((upper_volume(f,D3)|1)^(upper_volume(f,D3)/^1)) by RFINSEQ:8;
A586: D3.(len D3) = upper_bound A by INTEGRA1:def 2;
len D3 in dom D3 by A579,FINSEQ_1:def 3;
then
A587: len D3 > 1 by A580,A582,A586,A584,SEQ_4:137,XREAL_1:47;
then reconsider D=D3/^1 as increasing FinSequence of REAL by
INTEGRA1:34;
A588: len D = len D3 - 1 by A587,RFINSEQ:def 1;
upper_bound A > lower_bound A by A584,XREAL_1:47;
then len D <> 0 by A582,A588,INTEGRA1:def 2;
then reconsider D as non empty increasing FinSequence of REAL;
A589: len D in dom D by FINSEQ_5:6;
len D+1=len D3 by A588;
then
A590: D.(len D)=upper_bound A by A586,A587,A589,RFINSEQ:def 1;
A591: len D in Seg len D by FINSEQ_1:3;
1+1 <= len D3 by A587,NAT_1:13;
then 2 in dom D3 by FINSEQ_3:25;
then
A592: D3.1 < D3.2 by A580,SEQM_3:def 1;
A593: rng D3 c= A by INTEGRA1:def 2;
rng D c= rng D3 by FINSEQ_5:33;
then rng D c= A by A593;
then reconsider D as Division of A by A590,INTEGRA1:def 2;
A594: 1 in Seg 1 by FINSEQ_1:1;
A595: len D3 >= 1+1 by A587,NAT_1:13;
then
A596: 2 <= len upper_volume(f,D3) by INTEGRA1:def 6;
1 <= len upper_volume(f,D3) by A587,INTEGRA1:def 6;
then
A597: len(mid(upper_volume(f,D3),2,len upper_volume( f,D3))) =len
upper_volume(f,D3)-'2+1 by A596,FINSEQ_6:118
.=len D3-'2+1 by INTEGRA1:def 6
.=len D3-2+1 by A595,XREAL_1:233
.=len D3-1;
A598: for i be Nat st 1<=i&i<=len mid(upper_volume(f,D3),2,len
upper_volume(f,D3)) holds mid(upper_volume(f,D3),2,len upper_volume(f,D3)).i =
upper_volume(f,D).i
proof
let i be Nat;
assume that
A599: 1<=i and
A600: i<=len mid(upper_volume(f,D3),2,len upper_volume(f,D3));
A601: 1 <= i+1 by NAT_1:12;
i+1 <= len D3 by A597,A600,XREAL_1:19;
then
A602: i+1 in Seg len D3 by A601,FINSEQ_1:1;
then
A603: i+1 in dom D3 by FINSEQ_1:def 3;
A604: divset(D3,i+1)=divset(D,i)
proof
A605: i+1 in dom D3 by A602,FINSEQ_1:def 3;
A606: 1<>i+1 by A599,NAT_1:13;
then
A607: upper_bound divset(D3,i+1)=D3.(i+1) by A605,INTEGRA1:def 4;
A608: i in dom D by A588,A597,A599,A600,FINSEQ_3:25;
then
A609: D.i=D3.(i+1) by A587,RFINSEQ:def 1;
A610: lower_bound divset(D3,i+1)=D3.(i+1 -1) by A606,A605,
INTEGRA1:def 4;
per cases;
suppose
A611: i=1;
then
A612: upper_bound divset (D,i)=D.i by A608,INTEGRA1:def 4;
A613: lower_bound divset(D,i)=lower_bound A by A608,A611,
INTEGRA1:def 4;
divset(D3,i+1)=[.lower_bound A, D.i.] by A582,A607,A610,A609
,A611,INTEGRA1:4;
hence thesis by A613,A612,INTEGRA1:4;
end;
suppose
A614: i<>1;
then i-1 in dom D by A608,INTEGRA1:7;
then
A615: D.(i-1)=D3.(i-1+1) by A587,RFINSEQ:def 1
.=D3.i;
A616: upper_bound divset(D,i)=D.i by A608,A614,INTEGRA1:def 4;
lower_bound divset(D,i)=D.(i-1) by A608,A614,INTEGRA1:def 4;
then divset(D3,i+1)=[.lower_bound divset(D,i), upper_bound
divset(D,i).] by A607,A610,A609,A616,A615,INTEGRA1:4;
hence thesis by INTEGRA1:4;
end;
end;
i <= len upper_volume(f,D3) - 1 by A597,A600,INTEGRA1:def 6;
then
A617: i <= len upper_volume(f,D3)-2+1;
mid(upper_volume(f,D3),2,len upper_volume( f,D3)).i =
upper_volume(f,D3).(i+2-1) by A596,A599,A617,FINSEQ_6:122
.=upper_volume(f,D3).(i+1);
then
A618: mid(upper_volume(f,D3),2,len upper_volume( f,D3)).i =(
upper_bound rng(f|divset(D3,i+1)))*vol(divset(D3,i+1)) by A603,INTEGRA1:def 6;
i in Seg len D by A588,A597,A599,A600,FINSEQ_1:1;
then i in dom D by FINSEQ_1:def 3;
hence thesis by A618,A604,INTEGRA1:def 6;
end;
A619: 1 <= len upper_volume(f,D3) by A587,INTEGRA1:def 6;
A620: len (upper_volume(f,D3)|1)=1;
1 in dom upper_volume(f,D3) by A619,FINSEQ_3:25;
then (upper_volume(f,D3)|1).1 = upper_volume(f,D3).1 by A594,
RFINSEQ:6;
then
A621: upper_volume(f,D3)|1 = <*upper_volume(f,D3).1*> by A620,FINSEQ_1:40;
A622: 2-'1=2-1 by XREAL_1:233
.= 1;
1 <= len D by A591,FINSEQ_1:1;
then 1 in dom D by FINSEQ_3:25;
then
A623: D.1=D3.(1+1) by A587,RFINSEQ:def 1
.=D3.2;
D in divs A by INTEGRA1:def 3;
then
A624: D in dom upper_sum_set(f) by FUNCT_2:def 1;
len upper_volume(f,D3) >= 2 by A595,INTEGRA1:def 6;
then
A625: mid(upper_volume(f,D3),2,len upper_volume(f,D3 )) =
upper_volume(f,D3)/^1 by A622,FINSEQ_6:117;
len(mid(upper_volume(f,D3),2,len upper_volume( f,D3))) =len
upper_volume(f,D) by A588,A597,INTEGRA1:def 6;
then
A626: upper_volume(f,D3)/^1 = upper_volume(f,D) by A625,A598,FINSEQ_1:14;
vol (divset(D3,1))=upper_bound divset(D3,1)-lower_bound
divset(D3,1) by INTEGRA1:def 5
.=upper_bound divset(D3,1) - lower_bound A by A580,INTEGRA1:def 4
.=D3.1 - lower_bound A by A580,INTEGRA1:def 4
.=0 by A582;
then y=0+Sum(upper_volume(f,D)) by A585,A621,A583,A626,RVSUM_1:76
.=upper_sum(f,D) by INTEGRA1:def 8;
then y=(upper_sum_set(f)).D by INTEGRA1:def 10;
hence thesis by A582,A624,A623,A592;
end;
hence thesis;
end;
end;
then consider D being Division of A such that
D in dom upper_sum_set(f) and
A627: y=(upper_sum_set(f)).D and
A628: D.1>lower_bound A;
deffunc F(Nat)=In(vol(divset(D,$1)),REAL);
set p=len D, H=upper_bound rng f, h=lower_bound rng f;
consider v being FinSequence of REAL such that
A629: len v = len D & for j be Nat st j in dom v holds v.j=F(j) from
FINSEQ_2:sch 1;
A630: 2*p >0 by XREAL_1:129;
consider v1 being non-decreasing FinSequence of REAL such that
A631: v,v1 are_fiberwise_equipotent by INTEGRA2:3;
defpred P[Nat] means $1 in dom v1 & v1.$1 > 0;
A632: dom v = Seg len D by A629,FINSEQ_1:def 3;
A633: ex k be Nat st P[k]
proof
consider H being Function such that
dom H = dom v and
rng H = dom v1 and
H is one-to-one and
A634: v=v1*H by A631,CLASSES1:77;
consider k such that
A635: k in dom D and
A636: vol(divset(D,k)) > 0 by A562,Th2;
A637: dom D = Seg len D by FINSEQ_1:def 3;
then H.k in dom v1 by A632,A634,A635,FUNCT_1:11;
then reconsider Hk = H.k as Element of NAT;
v.k = F(k) by A629,A632,A635,A637;
then v.k > 0 by A636;
then P[Hk] by A632,A634,A635,A637,FUNCT_1:11,12;
hence thesis;
end;
consider k be Nat such that
A638: P[k] & for n be Nat st P[n] holds k<=n from NAT_1:sch 5(A633);
A639: H-h >= 0 by A1,Lm3,XREAL_1:48;
then
A640: 2*p*(H-h+1) > 0 by A630,XREAL_1:129;
min(v1.k,e/(2*p*(H-h+1))) > 0
proof
per cases by XXREAL_0:15;
suppose
min(v1.k,e/(2*p*(H-h+1))) = v1.k;
hence thesis by A638;
end;
suppose
min(v1.k,e/(2*p*(H-h+1))) = e/(2*p*(H-h+1));
hence thesis by A572,A640,XREAL_1:139;
end;
end;
then consider n being Nat such that
A641: for m being Nat st n<=m
holds 0 < (delta(T)).m & (delta(T)).m < min(v1.k,
e/(2*p*(H-h+1))) by A564;
take n;
A642: y=upper_sum(f,D) by A627,INTEGRA1:def 10;
A643: v1.1 > 0
proof
A644: for n1 be Element of NAT st n1 in dom D holds vol(divset(D,n1))
>0
proof
let n1 be Element of NAT;
assume
A645: n1 in dom D;
then
A646: 1 <= n1 by FINSEQ_3:25;
per cases by A646,XXREAL_0:1;
suppose
A647: n1=1;
then
A648: upper_bound divset(D,n1)=D.n1 by A645,INTEGRA1:def 4;
lower_bound divset(D,n1)=lower_bound A by A645,A647,INTEGRA1:def 4;
then vol(divset(D,n1))=D.n1-lower_bound A by A648,INTEGRA1:def 5;
hence thesis by A628,A647,XREAL_1:50;
end;
suppose
A649: n1>1;
then
A650: upper_bound divset(D,n1)=D.n1 by A645,INTEGRA1:def 4;
lower_bound divset(D,n1)=D.(n1-1) by A645,A649,INTEGRA1:def 4;
then
A651: vol(divset(D,n1))=D.n1-D.(n1-1) by A650,INTEGRA1:def 5;
n1 < n1+1 by XREAL_1:29;
then
A652: n1-1 < n1 by XREAL_1:19;
n1-1 in dom D by A645,A649,INTEGRA1:7;
then D.(n1-1)<D.n1 by A645,A652,SEQM_3:def 1;
hence thesis by A651,XREAL_1:50;
end;
end;
A653: k <= len v1 by A638,FINSEQ_3:25;
1 <= k by A638,FINSEQ_3:25;
then 1 <= len v1 by A653,XXREAL_0:2;
then 1 in dom v1 by FINSEQ_3:25;
then
A654: v1.1 in rng v1 by FUNCT_1:def 3;
rng v = rng v1 by A631,CLASSES1:75;
then consider n1 being Element of NAT such that
A655: n1 in dom v and
A656: v1.1 = v.n1 by A654,PARTFUN1:3;
n1 in Seg len D by A629,A655,FINSEQ_1:def 3;
then
A657: n1 in dom D by FINSEQ_1:def 3;
v1.1 = F(n1) by A629,A655,A656
.= vol(divset(D,n1));
hence thesis by A644,A657;
end;
A658: v1.k = min rng upper_volume(chi(A,A),D)
proof
A659: k=1
proof
len v1 = len v by A631,RFINSEQ:3;
then k in Seg len v by A638,FINSEQ_1:def 3;
then
A660: 1 <= k by FINSEQ_1:1;
k in Seg len v1 by A638,FINSEQ_1:def 3;
then k <= len v1 by FINSEQ_1:1;
then 1 <= len v1 by A660,XXREAL_0:2;
then
A661: 1 in dom v1 by FINSEQ_3:25;
assume k <> 1;
then k > 1 by A660,XXREAL_0:1;
hence contradiction by A638,A643,A661;
end;
A662: rng v = rng v1 by A631,CLASSES1:75;
v1.k in rng upper_volume(chi(A,A),D)
proof
v1.k in rng v by A638,A662,FUNCT_1:def 3;
then consider k2 being Element of NAT such that
A663: k2 in dom v and
A664: v1.k = v.k2 by PARTFUN1:3;
A665: k2 in Seg len D by A629,A663,FINSEQ_1:def 3;
then
A666: k2 in dom D by FINSEQ_1:def 3;
k2 in Seg len upper_volume(chi(A,A),D) by A665,INTEGRA1:def 6;
then
A667: k2 in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3;
v1.k = F(k2) by A629,A663,A664
.= vol(divset(D,k2));
then v1.k = upper_volume(chi(A,A),D).k2 by A666,INTEGRA1:20;
hence thesis by A667,FUNCT_1:def 3;
end;
then
A668: v1.k >= min rng upper_volume(chi(A,A),D) by XXREAL_2:def 7;
min rng upper_volume(chi(A,A),D) in rng upper_volume(chi(A,A),D
) by XXREAL_2:def 7;
then consider m such that
A669: m in dom upper_volume(chi(A,A),D) and
A670: min rng upper_volume(chi(A,A),D)=upper_volume(chi(A,A),D).m
by PARTFUN1:3;
m in Seg len upper_volume(chi(A,A),D) by A669,FINSEQ_1:def 3;
then
A671: m in Seg len D by INTEGRA1:def 6;
then m in dom D by FINSEQ_1:def 3;
then
A672: min rng upper_volume(chi(A,A),D)=vol(divset(D,m)) by A670,
INTEGRA1:20;
A673: v.m = F(m) by A629,A632,A671
.=min rng upper_volume(chi(A,A),D) by A672;
m in dom v by A629,A671,FINSEQ_1:def 3;
then min rng upper_volume(chi(A,A),D) in rng v by A673,FUNCT_1:def 3;
then consider m1 being Element of NAT such that
A674: m1 in dom v1 and
A675: min rng upper_volume(chi(A,A),D)=v1.m1 by A662,PARTFUN1:3;
m1 >= 1 by A674,FINSEQ_3:25;
then v1.1 <= min rng upper_volume(chi(A,A),D) by A638,A659,A674,A675,
INTEGRA2:2;
hence thesis by A659,A668,XXREAL_0:1;
end;
A676: min (v1.k,e/(2*p*(H-h+1))) <= v1.k by XXREAL_0:17;
set s=upper_integral(f), sD=upper_sum(f,D);
let m be Nat;
reconsider mm=m as Element of NAT by ORDINAL1:def 12;
reconsider D1 = T.mm as Division of A;
A677: delta(D1)=(delta(T)).m by Def2;
consider D2 being Division of A such that
A678: D <= D2 and
D1 <= D2 and
A679: rng D2 = rng D1 \/ rng D and
0 <= upper_sum(f,D)-upper_sum(f,D2) and
0 <= upper_sum(f,D1)-upper_sum(f,D2) by A2;
set sD1=upper_sum(f,T.mm), sD2=upper_sum(f,D2);
upper_sum(f,D2) <= upper_sum(f,D) by A1,A678,INTEGRA1:45;
then
A680: sD1-sD <= sD1-sD2 by XREAL_1:10;
sD+sD1-sD1-s < e/2 by A576,A642,XREAL_1:19;
then sD1-s+sD-sD1 < e/2;
then sD1-s+sD < sD1+e/2 by XREAL_1:19;
then
A681: sD1-s < sD1+e/2-sD by XREAL_1:20;
T.mm in divs A by INTEGRA1:def 3;
then
A682: T.m in dom upper_sum_set(f) by FUNCT_2:def 1;
(upper_sum(f,T)).m = upper_sum(f,T.mm) by INTEGRA2:def 2;
then (upper_sum(f,T)).m = (upper_sum_set(f)).(T.m) by INTEGRA1:def 10;
then (upper_sum(f,T)).m in rng upper_sum_set(f) by A682,FUNCT_1:def 3;
then lower_bound rng upper_sum_set(f)<=(upper_sum(f,T)).m by A574,
SEQ_4:def 2;
then upper_integral(f)<=(upper_sum(f,T)).m by INTEGRA1:def 14;
then
A683: (upper_sum(f,T)).m-upper_integral(f) >= 0 by XREAL_1:48;
H-h <= H-h+1 by XREAL_1:29;
then
A684: p*(H-h)<=p*(H-h+1) by XREAL_1:64;
A685: min(v1.k,e/(2*p*(H-h+1))) <= e/(2*p*(H-h+1)) by XXREAL_0:17;
assume
A686: n<=m;
then (delta(T)).m < min(v1.k,e/(2*p*(H-h+1))) by A641;
then (delta(T)).m < e/(2*p*(H-h+1)) by A685,XXREAL_0:2;
then (delta(T)).m*(2*p*(H-h+1))<e by A630,A639,XREAL_1:79,129;
then ((delta(T)).m*(p*(H-h+1)))*2<e;
then
A687: p*(H-h+1)*(delta(T)).m < e/2 by XREAL_1:81;
(delta(T)).m < min(v1.k,e/(2*p*(H-h+1))) by A641,A686;
then delta(D1)<v1.k by A677,A676,XXREAL_0:2;
then ex D3 being Division of A st D<=D3 & D1<=D3 & rng D3= rng D1
\/ rng D & upper_sum(f,D1)-upper_sum(f,D3) <=(len D)*(upper_bound rng f -
lower_bound rng f)*delta(D1) by A7,A658;
then
A688: upper_sum(f,D1)-upper_sum(f,D2)<= (len D)*(upper_bound rng f-
lower_bound rng f)*delta(D1) by A679,Th6;
0 < (delta(T)).m by A641,A686;
then p*(H-h)*(delta(T)).m <= p*(H-h+1)*(delta(T)).m by A684,XREAL_1:64;
then upper_sum(f,T.mm)-upper_sum(f,D2)
<= p*(H-h+1)*(delta(T)).m by A677
,A688,XXREAL_0:2;
then sD1-sD <= p*(H-h+1)*(delta(T)).m by A680,XXREAL_0:2;
then sD1-sD < e/2 by A687,XXREAL_0:2;
then sD1-sD+e/2 < e/2+e/2 by XREAL_1:6;
then sD1-s < e by A681,XXREAL_0:2;
then (upper_sum(f,T)).m-upper_integral(f) < e by INTEGRA2:def 2;
hence thesis by A683,ABSVALUE:def 1;
end;
hence upper_sum(f,T) is convergent by SEQ_2:def 6;
hence thesis by A571,SEQ_2:def 7;
end;