:: INTEGRA3 semantic presentation
begin
Lm1: for j being Element of NAT holds (j -' j) + 1 = 1
proof
let j be Element of NAT ; ::_thesis: (j -' j) + 1 = 1
j -' j = j - j by XREAL_1:233
.= 0 ;
hence (j -' j) + 1 = 1 ; ::_thesis: verum
end;
Lm2: for n being Element of NAT st 1 <= n & n <= 2 & not n = 1 holds
n = 2
proof
let n be Element of NAT ; ::_thesis: ( 1 <= n & n <= 2 & not n = 1 implies n = 2 )
assume that
A1: 1 <= n and
A2: n <= 2 ; ::_thesis: ( n = 1 or n = 2 )
percases ( n = 1 or n > 1 ) by A1, XXREAL_0:1;
suppose n = 1 ; ::_thesis: ( n = 1 or n = 2 )
hence ( n = 1 or n = 2 ) ; ::_thesis: verum
end;
suppose n > 1 ; ::_thesis: ( n = 1 or n = 2 )
then n >= 1 + 1 by NAT_1:13;
hence ( n = 1 or n = 2 ) by A2, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
definition
let A be non empty closed_interval Subset of REAL;
let D be Division of A;
func delta D -> Real equals :: INTEGRA3:def 1
max (rng (upper_volume ((chi (A,A)),D)));
correctness
coherence
max (rng (upper_volume ((chi (A,A)),D))) is Real;
by XREAL_0:def_1;
end;
:: deftheorem defines delta INTEGRA3:def_1_:_
for A being non empty closed_interval Subset of REAL
for D being Division of A holds delta D = max (rng (upper_volume ((chi (A,A)),D)));
definition
let A be non empty closed_interval Subset of REAL;
let T be DivSequence of A;
func delta T -> Real_Sequence means :Def2: :: INTEGRA3:def 2
for i being Element of NAT holds it . i = delta (T . i);
existence
ex b1 being Real_Sequence st
for i being Element of NAT holds b1 . i = delta (T . i)
proof
deffunc H1( Element of NAT ) -> Real = delta (T . $1);
thus ex IT being Real_Sequence st
for i being Element of NAT holds IT . i = H1(i) from SEQ_1:sch_1(); ::_thesis: verum
end;
uniqueness
for b1, b2 being Real_Sequence st ( for i being Element of NAT holds b1 . i = delta (T . i) ) & ( for i being Element of NAT holds b2 . i = delta (T . i) ) holds
b1 = b2
proof
let F1, F2 be Real_Sequence; ::_thesis: ( ( for i being Element of NAT holds F1 . i = delta (T . i) ) & ( for i being Element of NAT holds F2 . i = delta (T . i) ) implies F1 = F2 )
assume that
A1: for i being Element of NAT holds F1 . i = delta (T . i) and
A2: for i being Element of NAT holds F2 . i = delta (T . i) ; ::_thesis: F1 = F2
for i being Element of NAT holds F1 . i = F2 . i
proof
let i be Element of NAT ; ::_thesis: F1 . i = F2 . i
F1 . i = delta (T . i) by A1
.= F2 . i by A2 ;
hence F1 . i = F2 . i ; ::_thesis: verum
end;
hence F1 = F2 by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def2 defines delta INTEGRA3:def_2_:_
for A being non empty closed_interval Subset of REAL
for T being DivSequence of A
for b3 being Real_Sequence holds
( b3 = delta T iff for i being Element of NAT holds b3 . i = delta (T . i) );
theorem :: INTEGRA3:1
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A st D1 <= D2 holds
delta D1 >= delta D2
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A st D1 <= D2 holds
delta D1 >= delta D2
let D1, D2 be Division of A; ::_thesis: ( D1 <= D2 implies delta D1 >= delta D2 )
delta D2 in rng (upper_volume ((chi (A,A)),D2)) by XXREAL_2:def_8;
then consider j being Element of NAT 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 ; ::_thesis: delta D1 >= delta D2
then consider i being Element of 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 delta D1 >= delta D2 ; ::_thesis: verum
end;
theorem Th2: :: INTEGRA3:2
for A being non empty closed_interval Subset of REAL
for D being Division of A st vol A <> 0 holds
ex i being Element of NAT st
( i in dom D & vol (divset (D,i)) > 0 )
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A st vol A <> 0 holds
ex i being Element of NAT st
( i in dom D & vol (divset (D,i)) > 0 )
let D be Division of A; ::_thesis: ( vol A <> 0 implies ex i being Element of NAT st
( i in dom D & vol (divset (D,i)) > 0 ) )
assume A1: vol A <> 0 ; ::_thesis: ex i being Element of NAT st
( i in dom D & vol (divset (D,i)) > 0 )
A2: len D in dom D by FINSEQ_5:6;
assume A3: for i being Element of NAT st i in dom D holds
vol (divset (D,i)) <= 0 ; ::_thesis: contradiction
A4: for i being Element of NAT st i in dom D holds
vol (divset (D,i)) = 0
proof
let i be Element of NAT ; ::_thesis: ( i in dom D implies vol (divset (D,i)) = 0 )
assume i in dom D ; ::_thesis: vol (divset (D,i)) = 0
then vol (divset (D,i)) <= 0 by A3;
hence vol (divset (D,i)) = 0 by INTEGRA1:9; ::_thesis: verum
end;
A5: for i being Element of NAT st i in dom D holds
upper_bound (divset (D,i)) = lower_bound (divset (D,i))
proof
let i be Element of NAT ; ::_thesis: ( i in dom D implies upper_bound (divset (D,i)) = lower_bound (divset (D,i)) )
assume i in dom D ; ::_thesis: upper_bound (divset (D,i)) = lower_bound (divset (D,i))
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 upper_bound (divset (D,i)) = lower_bound (divset (D,i)) ; ::_thesis: verum
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 ; ::_thesis: contradiction
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; ::_thesis: verum
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; ::_thesis: verum
end;
theorem Th3: :: INTEGRA3:3
for x being Real
for A being non empty closed_interval Subset of REAL
for D being Division of A st x in A holds
ex j being Element of NAT st
( j in dom D & x in divset (D,j) )
proof
let x be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A st x in A holds
ex j being Element of NAT st
( j in dom D & x in divset (D,j) )
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A st x in A holds
ex j being Element of NAT st
( j in dom D & x in divset (D,j) )
let D be Division of A; ::_thesis: ( x in A implies ex j being Element of NAT st
( j in dom D & x in divset (D,j) ) )
assume A1: x in A ; ::_thesis: ex j being Element of NAT st
( j in dom D & x in divset (D,j) )
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;
percases ( x in rng D or not x in rng D ) ;
suppose x in rng D ; ::_thesis: ex j being Element of NAT st
( j in dom D & x in divset (D,j) )
then consider j being Element of NAT such that
A6: j in dom D and
A7: D . j = x by PARTFUN1:3;
x in divset (D,j)
proof
percases ( j = 1 or j <> 1 ) ;
supposeA8: j = 1 ; ::_thesis: x in divset (D,j)
A9: ex a, b being Real 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 x in divset (D,j) by A7, A9, INTEGRA2:1; ::_thesis: verum
end;
supposeA10: j <> 1 ; ::_thesis: x in divset (D,j)
A11: ex a, b being Real 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 x in divset (D,j) by A7, A11, INTEGRA2:1; ::_thesis: verum
end;
end;
end;
hence ex j being Element of NAT st
( j in dom D & x in divset (D,j) ) by A6; ::_thesis: verum
end;
supposeA12: not x in rng D ; ::_thesis: ex j being Element of NAT st
( j in dom D & x in divset (D,j) )
defpred S1[ 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
percases ( len D = 1 or len D <> 1 ) ;
suppose len D = 1 ; ::_thesis: upper_bound (divset (D,(len D))) = D . (len D)
hence upper_bound (divset (D,(len D))) = D . (len D) by A13, INTEGRA1:def_4; ::_thesis: verum
end;
suppose len D <> 1 ; ::_thesis: upper_bound (divset (D,(len D))) = D . (len D)
hence upper_bound (divset (D,(len D))) = D . (len D) by A13, INTEGRA1:def_4; ::_thesis: verum
end;
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 ; ::_thesis: contradiction
then x = D . (len D) by INTEGRA1:def_2;
hence contradiction by A12, A13, FUNCT_1:def_3; ::_thesis: verum
end;
then x < upper_bound (divset (D,(len D))) by A3, A14, XXREAL_0:1;
then A15: ex k being Nat st S1[k] by A13;
consider k being Nat such that
A16: ( S1[k] & ( for n being Nat st S1[n] holds
k <= n ) ) from NAT_1:sch_5(A15);
defpred S2[ 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 being Nat st S2[k] by A2, A4, FINSEQ_3:32;
A18: for k being Nat st S2[k] holds
k <= len D by FINSEQ_3:25;
consider j being Nat such that
A19: ( S2[j] & ( for n being Nat st S2[n] holds
n <= j ) ) from NAT_1:sch_6(A18, A17);
k = j
proof
assume A20: k <> j ; ::_thesis: contradiction
percases ( k < j or k > j ) by A20, XXREAL_0:1;
supposeA21: k < j ; ::_thesis: contradiction
A22: upper_bound (divset (D,k)) = D . k
proof
percases ( k = 1 or k <> 1 ) ;
suppose k = 1 ; ::_thesis: upper_bound (divset (D,k)) = D . k
hence upper_bound (divset (D,k)) = D . k by A16, INTEGRA1:def_4; ::_thesis: verum
end;
suppose k <> 1 ; ::_thesis: upper_bound (divset (D,k)) = D . k
hence upper_bound (divset (D,k)) = D . k by A16, INTEGRA1:def_4; ::_thesis: verum
end;
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; ::_thesis: verum
end;
supposeA25: k > j ; ::_thesis: contradiction
x < upper_bound (divset (D,j))
proof
A26: upper_bound (divset (D,j)) = D . j
proof
percases ( j = 1 or j <> 1 ) ;
suppose j = 1 ; ::_thesis: upper_bound (divset (D,j)) = D . j
hence upper_bound (divset (D,j)) = D . j by A19, INTEGRA1:def_4; ::_thesis: verum
end;
suppose j <> 1 ; ::_thesis: upper_bound (divset (D,j)) = D . j
hence upper_bound (divset (D,j)) = D . j by A19, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
assume A27: x >= upper_bound (divset (D,j)) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
hence contradiction by A16, A19, A25; ::_thesis: verum
end;
end;
end;
then x in divset (D,k) by A16, A19, INTEGRA2:1;
hence ex j being Element of NAT st
( j in dom D & x in divset (D,j) ) by A16; ::_thesis: verum
end;
end;
end;
theorem Th4: :: INTEGRA3:4
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A ex D being Division of A st
( D1 <= D & D2 <= D & rng D = (rng D1) \/ (rng D2) )
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A ex D being Division of A st
( D1 <= D & D2 <= D & rng D = (rng D1) \/ (rng D2) )
let D1, D2 be Division of A; ::_thesis: ex D being Division of A st
( D1 <= D & D2 <= D & rng D = (rng D1) \/ (rng D2) )
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 = D as increasing FinSequence of REAL by A3;
reconsider D = 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 being Element of NAT 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 ; ::_thesis: contradiction
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; ::_thesis: verum
end;
then reconsider D = D as Division of A by A7, INTEGRA1:def_2;
take D ; ::_thesis: ( D1 <= D & D2 <= D & rng D = (rng D1) \/ (rng D2) )
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; ::_thesis: ( D2 <= D & rng D = (rng D1) \/ (rng D2) )
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; ::_thesis: rng D = (rng D1) \/ (rng D2)
thus rng D = (rng D1) \/ (rng D2) by A1, FINSEQ_1:31; ::_thesis: verum
end;
theorem Th5: :: INTEGRA3:5
for A being non empty closed_interval Subset of REAL
for D1, D being Division of A st delta D1 < min (rng (upper_volume ((chi (A,A)),D))) holds
for x, y being Real
for i being Element of NAT st i in dom D1 & x in (rng D) /\ (divset (D1,i)) & y in (rng D) /\ (divset (D1,i)) holds
x = y
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D being Division of A st delta D1 < min (rng (upper_volume ((chi (A,A)),D))) holds
for x, y being Real
for i being Element of NAT st i in dom D1 & x in (rng D) /\ (divset (D1,i)) & y in (rng D) /\ (divset (D1,i)) holds
x = y
let D1, D be Division of A; ::_thesis: ( delta D1 < min (rng (upper_volume ((chi (A,A)),D))) implies for x, y being Real
for i being Element of NAT st i in dom D1 & x in (rng D) /\ (divset (D1,i)) & y in (rng D) /\ (divset (D1,i)) holds
x = y )
assume A1: delta D1 < min (rng (upper_volume ((chi (A,A)),D))) ; ::_thesis: for x, y being Real
for i being Element of NAT st i in dom D1 & x in (rng D) /\ (divset (D1,i)) & y in (rng D) /\ (divset (D1,i)) holds
x = y
let x, y be Real; ::_thesis: for i being Element of NAT st i in dom D1 & x in (rng D) /\ (divset (D1,i)) & y in (rng D) /\ (divset (D1,i)) holds
x = y
let i be Element of NAT ; ::_thesis: ( i in dom D1 & x in (rng D) /\ (divset (D1,i)) & y in (rng D) /\ (divset (D1,i)) implies x = y )
assume A2: i in dom D1 ; ::_thesis: ( not x in (rng D) /\ (divset (D1,i)) or not y in (rng D) /\ (divset (D1,i)) or x = y )
assume A3: x in (rng D) /\ (divset (D1,i)) ; ::_thesis: ( not y in (rng D) /\ (divset (D1,i)) or x = y )
then x in rng D by XBOOLE_0:def_4;
then consider n being Element of NAT such that
A4: n in dom D and
A5: x = D . n by PARTFUN1:3;
assume A6: y in (rng D) /\ (divset (D1,i)) ; ::_thesis: x = y
then y in rng D by XBOOLE_0:def_4;
then consider m being Element of NAT such that
A7: m in dom D and
A8: y = D . m by PARTFUN1:3;
assume A9: x <> y ; ::_thesis: contradiction
A10: abs ((D . n) - (D . m)) >= min (rng (upper_volume ((chi (A,A)),D)))
proof
percases ( n < m or n > m ) by A9, A5, A8, XXREAL_0:1;
suppose n < m ; ::_thesis: abs ((D . n) - (D . m)) >= min (rng (upper_volume ((chi (A,A)),D)))
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;
- (abs ((D . n) - (D . m))) <= (D . n) - (D . m) by ABSVALUE:4;
then A19: abs ((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 abs ((D . n) - (D . m)) >= min (rng (upper_volume ((chi (A,A)),D))) by A19, XXREAL_0:2; ::_thesis: verum
end;
suppose n > m ; ::_thesis: abs ((D . n) - (D . m)) >= min (rng (upper_volume ((chi (A,A)),D)))
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;
abs ((D . n) - (D . m)) >= (D . n) - (D . m) by ABSVALUE:4;
hence abs ((D . n) - (D . m)) >= min (rng (upper_volume ((chi (A,A)),D))) by A28, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
abs ((D . n) - (D . m)) <= delta D1
proof
percases ( n < m or n > m ) by A9, A5, A8, XXREAL_0:1;
supposeA29: n < m ; ::_thesis: abs ((D . n) - (D . m)) <= delta D1
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 abs ((D . n) - (D . m)) = - ((D . n) - (D . m)) by ABSVALUE:def_1
.= (D . m) - (D . n) ;
hence abs ((D . n) - (D . m)) <= delta D1 by A32, A30, XXREAL_0:2; ::_thesis: verum
end;
supposeA33: n > m ; ::_thesis: abs ((D . n) - (D . m)) <= delta D1
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 abs ((D . n) - (D . m)) = (D . n) - (D . m) by ABSVALUE:def_1;
hence abs ((D . n) - (D . m)) <= delta D1 by A36, A34, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence contradiction by A1, A10, XXREAL_0:2; ::_thesis: verum
end;
theorem Th6: :: INTEGRA3:6
for p, q being FinSequence of REAL st rng p = rng q & p is increasing & q is increasing holds
p = q
proof
let p, q be FinSequence of REAL ; ::_thesis: ( rng p = rng q & p is increasing & q is increasing implies p = q )
assume A1: rng p = rng q ; ::_thesis: ( not p is increasing or not q is increasing or p = q )
assume that
A2: p is increasing and
A3: q is increasing ; ::_thesis: p = q
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 p = q by A1, A2, A3, SEQ_4:141; ::_thesis: verum
end;
theorem Th7: :: INTEGRA3:7
for i, j being Element of NAT
for A being non empty closed_interval Subset of REAL
for D, D1 being Division of A st D <= D1 & i in dom D & j in dom D & i <= j holds
( indx (D1,D,i) <= indx (D1,D,j) & indx (D1,D,i) in dom D1 )
proof
let i, j be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D, D1 being Division of A st D <= D1 & i in dom D & j in dom D & i <= j holds
( indx (D1,D,i) <= indx (D1,D,j) & indx (D1,D,i) in dom D1 )
let A be non empty closed_interval Subset of REAL; ::_thesis: for D, D1 being Division of A st D <= D1 & i in dom D & j in dom D & i <= j holds
( indx (D1,D,i) <= indx (D1,D,j) & indx (D1,D,i) in dom D1 )
let D, D1 be Division of A; ::_thesis: ( 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 ) )
assume that
A1: D <= D1 and
A2: i in dom D and
A3: j in dom D and
A4: i <= j ; ::_thesis: ( indx (D1,D,i) <= indx (D1,D,j) & indx (D1,D,i) in dom D1 )
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 ( indx (D1,D,i) <= indx (D1,D,j) & indx (D1,D,i) in dom D1 ) by A5, A8, A7, A6, SEQM_3:def_1; ::_thesis: verum
end;
theorem Th8: :: INTEGRA3:8
for i, j being Element of NAT
for A being non empty closed_interval Subset of REAL
for D, D1 being Division of A st D <= D1 & i in dom D & j in dom D & i < j holds
indx (D1,D,i) < indx (D1,D,j)
proof
let i, j be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D, D1 being Division of A st D <= D1 & i in dom D & j in dom D & i < j holds
indx (D1,D,i) < indx (D1,D,j)
let A be non empty closed_interval Subset of REAL; ::_thesis: for D, D1 being Division of A st D <= D1 & i in dom D & j in dom D & i < j holds
indx (D1,D,i) < indx (D1,D,j)
let D, D1 be Division of A; ::_thesis: ( D <= D1 & i in dom D & j in dom D & i < j implies indx (D1,D,i) < indx (D1,D,j) )
assume that
A1: D <= D1 and
A2: i in dom D and
A3: j in dom D and
A4: i < j ; ::_thesis: indx (D1,D,i) < indx (D1,D,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 indx (D1,D,i) < indx (D1,D,j) by A5, A8, A7, A6, SEQ_4:137; ::_thesis: verum
end;
theorem Th9: :: INTEGRA3:9
for A being non empty closed_interval Subset of REAL
for D being Division of A holds delta D >= 0
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A holds delta D >= 0
let D be Division of A; ::_thesis: delta D >= 0
consider y being Real such that
A1: y in rng D by SUBSET_1:4;
consider n being Element of NAT 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 delta D >= 0 by A3; ::_thesis: verum
end;
Lm3: for A being non empty closed_interval Subset of REAL
for g being 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; ::_thesis: for g being Function of A,REAL st g | A is bounded holds
upper_bound (rng g) >= lower_bound (rng g)
let g be Function of A,REAL; ::_thesis: ( g | A is bounded implies upper_bound (rng g) >= lower_bound (rng g) )
assume A1: g | A is bounded ; ::_thesis: upper_bound (rng g) >= lower_bound (rng g)
then A2: rng g is bounded_below by INTEGRA1:11;
rng g is bounded_above by A1, INTEGRA1:13;
hence upper_bound (rng g) >= lower_bound (rng g) by A2, SEQ_4:11; ::_thesis: verum
end;
Lm4: for A, B being non empty closed_interval Subset of REAL
for f being 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; ::_thesis: for f being 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) )
let f be Function of A,REAL; ::_thesis: ( f | A is bounded & B c= A implies ( 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) ) )
assume that
A1: f | A is bounded and
A2: B c= A ; ::_thesis: ( 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) )
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 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; ::_thesis: ( 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) )
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; ::_thesis: ( upper_bound (rng (f | B)) <= upper_bound (rng f) & lower_bound (rng (f | B)) <= upper_bound (rng f) )
thus upper_bound (rng (f | B)) <= upper_bound (rng f) by A10, A4, RELAT_1:70, SEQ_4:48; ::_thesis: lower_bound (rng (f | B)) <= upper_bound (rng f)
hence lower_bound (rng (f | B)) <= upper_bound (rng f) by A11, XXREAL_0:2; ::_thesis: verum
end;
Lm5: for j being Element of NAT
for A being non empty closed_interval Subset of REAL
for D1 being Division of A st j in dom D1 holds
vol (divset (D1,j)) <= delta D1
proof
let j be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D1 being Division of A st j in dom D1 holds
vol (divset (D1,j)) <= delta D1
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1 being Division of A st j in dom D1 holds
vol (divset (D1,j)) <= delta D1
let D1 be Division of A; ::_thesis: ( j in dom D1 implies vol (divset (D1,j)) <= delta D1 )
assume A1: j in dom D1 ; ::_thesis: vol (divset (D1,j)) <= delta 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 vol (divset (D1,j)) <= delta D1 ; ::_thesis: verum
end;
Lm6: for x being Real
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for j1 being 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 x be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for j1 being 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)
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for j1 being 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)
let D1, D2 be Division of A; ::_thesis: for j1 being 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)
let j1 be Element of NAT ; ::_thesis: ( j1 = (len D1) - 1 & x in divset (D1,(len D1)) & len D1 >= 2 & D1 <= D2 & rng D2 = (rng D1) \/ {x} implies rng (D2 | (indx (D2,D1,j1))) = rng (D1 | j1) )
assume that
A1: j1 = (len D1) - 1 and
A2: x in divset (D1,(len D1)) and
A3: len D1 >= 2 ; ::_thesis: ( not D1 <= D2 or not rng D2 = (rng D1) \/ {x} or rng (D2 | (indx (D2,D1,j1))) = rng (D1 | j1) )
A4: len D1 in dom D1 by FINSEQ_5:6;
assume that
A5: D1 <= D2 and
A6: rng D2 = (rng D1) \/ {x} ; ::_thesis: rng (D2 | (indx (D2,D1,j1))) = rng (D1 | j1)
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 set st x1 in rng (D2 | (indx (D2,D1,j1))) holds
x1 in rng (D1 | j1)
proof
let x1 be set ; ::_thesis: ( x1 in rng (D2 | (indx (D2,D1,j1))) implies x1 in rng (D1 | j1) )
assume x1 in rng (D2 | (indx (D2,D1,j1))) ; ::_thesis: x1 in rng (D1 | j1)
then consider k being Element of NAT 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 ; ::_thesis: D2 . k in rng (D1 | j1)
then consider m being Element of NAT 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 D2 . k in rng (D1 | j1) by A26, FUNCT_1:def_3; ::_thesis: verum
end;
A27: ( D2 . k in {x} implies D2 . k = D1 . j1 )
proof
assume D2 . k in {x} ; ::_thesis: D2 . k = D1 . j1
then D1 . j1 <= D2 . k by A14, TARSKI:def_1;
hence D2 . k = D1 . j1 by A20, XXREAL_0:1; ::_thesis: verum
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} ; ::_thesis: D2 . k in rng (D1 | j1)
j1 in Seg j1 by A12, FINSEQ_1:1;
hence D2 . k in rng (D1 | j1) by A11, A27, A30, A29, RFINSEQ:6; ::_thesis: verum
end;
D2 . k in rng D2 by A18, FUNCT_1:def_3;
hence x1 in rng (D1 | j1) by A6, A9, A16, A17, A28, A21, RFINSEQ:6, XBOOLE_0:def_3; ::_thesis: verum
end;
then A31: rng (D2 | (indx (D2,D1,j1))) c= rng (D1 | j1) by TARSKI:def_3;
for x1 being set st x1 in rng (D1 | j1) holds
x1 in rng (D2 | (indx (D2,D1,j1)))
proof
let x1 be set ; ::_thesis: ( x1 in rng (D1 | j1) implies x1 in rng (D2 | (indx (D2,D1,j1))) )
assume x1 in rng (D1 | j1) ; ::_thesis: x1 in rng (D2 | (indx (D2,D1,j1)))
then consider k being Element of NAT 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 being Element of NAT 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 x1 in rng (D2 | (indx (D2,D1,j1))) by A39, A42, FUNCT_1:def_3; ::_thesis: verum
end;
then rng (D1 | j1) c= rng (D2 | (indx (D2,D1,j1))) by TARSKI:def_3;
hence rng (D2 | (indx (D2,D1,j1))) = rng (D1 | j1) by A31, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th10: :: INTEGRA3:10
for x being Real
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for g being Function of A,REAL st x in divset (D1,(len D1)) & len D1 >= 2 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & g | A is bounded holds
(Sum (lower_volume (g,D2))) - (Sum (lower_volume (g,D1))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1)
proof
let x be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for g being Function of A,REAL st x in divset (D1,(len D1)) & len D1 >= 2 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & g | A is bounded holds
(Sum (lower_volume (g,D2))) - (Sum (lower_volume (g,D1))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1)
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for g being Function of A,REAL st x in divset (D1,(len D1)) & len D1 >= 2 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & g | A is bounded holds
(Sum (lower_volume (g,D2))) - (Sum (lower_volume (g,D1))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1)
let D1, D2 be Division of A; ::_thesis: for g being Function of A,REAL st x in divset (D1,(len D1)) & len D1 >= 2 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & g | A is bounded holds
(Sum (lower_volume (g,D2))) - (Sum (lower_volume (g,D1))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1)
let g be Function of A,REAL; ::_thesis: ( 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) )
assume that
A1: x in divset (D1,(len D1)) and
A2: len D1 >= 2 ; ::_thesis: ( not D1 <= D2 or not rng D2 = (rng D1) \/ {x} or not g | A is bounded or (Sum (lower_volume (g,D2))) - (Sum (lower_volume (g,D1))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1) )
set j = len D1;
assume that
A3: D1 <= D2 and
A4: rng D2 = (rng D1) \/ {x} ; ::_thesis: ( not g | A is bounded or (Sum (lower_volume (g,D2))) - (Sum (lower_volume (g,D1))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1) )
A5: len D1 in dom D1 by FINSEQ_5:6;
then A6: indx (D2,D1,(len D1)) 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, INTEGRA1:35;
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 being Element of NAT st 1 <= k & k <= j1 holds
k = indx (D2,D1,k)
proof
let k be Element of NAT ; ::_thesis: ( 1 <= k & k <= j1 implies k = indx (D2,D1,k) )
assume that
A21: 1 <= k and
A22: k <= j1 ; ::_thesis: k = indx (D2,D1,k)
assume A23: k <> indx (D2,D1,k) ; ::_thesis: contradiction
percases ( k > indx (D2,D1,k) or k < indx (D2,D1,k) ) by A23, XXREAL_0:1;
supposeA24: k > indx (D2,D1,k) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
supposeA31: k < indx (D2,D1,k) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
end;
end;
A38: for k being 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; ::_thesis: ( 1 <= k & k <= len ((lower_volume (g,D1)) | j1) implies ((lower_volume (g,D1)) | j1) . k = ((lower_volume (g,D2)) | (indx (D2,D1,j1))) . k )
assume that
A40: 1 <= k and
A41: k <= len ((lower_volume (g,D1)) | j1) ; ::_thesis: ((lower_volume (g,D1)) | j1) . k = ((lower_volume (g,D2)) | (indx (D2,D1,j1))) . k
reconsider k = 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
percases ( k = 1 or k <> 1 ) ;
supposeA51: k = 1 ; ::_thesis: ( 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)))) )
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 ( 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)))) ) by A45, A49, A53, A52, INTEGRA1:def_4; ::_thesis: verum
end;
supposeA54: k <> 1 ; ::_thesis: ( 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)))) )
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 ( 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)))) ) by A3, A44, A61, A60, A55, A59, A62, INTEGRA1:def_19; ::_thesis: verum
end;
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 ((lower_volume (g,D1)) | j1) . k = ((lower_volume (g,D2)) | (indx (D2,D1,j1))) . k by A64; ::_thesis: verum
end;
A66: len D2 in dom D2 by FINSEQ_5:6;
deffunc H1( Division of A) -> FinSequence of REAL = lower_volume (g,$1);
deffunc H2( Division of A, Nat) -> Element of REAL = (PartSums (lower_volume (g,$1))) . $2;
A67: len D1 >= len (lower_volume (g,D1)) by INTEGRA1:def_7;
A68: len D1 <= len H1(D1) by INTEGRA1:def_7;
A69: len D1 in Seg (len D1) by FINSEQ_1:3;
then A70: 1 <= len D1 by FINSEQ_1:1;
then A71: len D1 in dom H1(D1) by A68, FINSEQ_3:25;
assume A72: g | A is bounded ; ::_thesis: (Sum (lower_volume (g,D2))) - (Sum (lower_volume (g,D1))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1)
len D1 < (len D1) + 1 by NAT_1:13;
then A73: j1 < len D1 by XREAL_1:19;
then j1 < len H1(D1) by INTEGRA1:def_7;
then j1 in dom H1(D1) by A10, FINSEQ_3:25;
then H2(D1,j1) = Sum (H1(D1) | j1) by INTEGRA1:def_20;
then H2(D1,j1) + (Sum (mid (H1(D1),(len D1),(len D1)))) = Sum ((H1(D1) | j1) ^ (mid (H1(D1),(len D1),(len D1)))) by RVSUM_1:75
.= Sum ((mid (H1(D1),1,j1)) ^ (mid (H1(D1),(j1 + 1),(len D1)))) by A10, FINSEQ_6:116
.= Sum (mid (H1(D1),1,(len D1))) by A10, A68, A73, INTEGRA2:4
.= Sum (H1(D1) | (len D1)) by A70, FINSEQ_6:116 ;
then A74: H2(D1,j1) + (Sum (mid ((lower_volume (g,D1)),(len D1),(len D1)))) = H2(D1, len D1) by A71, INTEGRA1:def_20;
A75: indx (D2,D1,(len D1)) in dom D2 by A3, A5, INTEGRA1:def_19;
then A76: indx (D2,D1,(len D1)) in Seg (len D2) by FINSEQ_1:def_3;
then A77: 1 <= indx (D2,D1,(len D1)) 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,(len D1))) - (indx (D2,D1,j1)) <= 2
proof
set ID1 = (indx (D2,D1,j1)) + 1;
set ID2 = ((indx (D2,D1,j1)) + 1) + 1;
assume (indx (D2,D1,(len D1))) - (indx (D2,D1,j1)) > 2 ; ::_thesis: contradiction
then A83: (indx (D2,D1,j1)) + (1 + 1) < indx (D2,D1,(len D1)) by XREAL_1:20;
A84: (indx (D2,D1,j1)) + 1 < ((indx (D2,D1,j1)) + 1) + 1 by NAT_1:13;
then indx (D2,D1,j1) <= ((indx (D2,D1,j1)) + 1) + 1 by NAT_1:13;
then A85: 1 <= ((indx (D2,D1,j1)) + 1) + 1 by A14, XXREAL_0:2;
A86: indx (D2,D1,(len D1)) in dom D2 by A3, A5, INTEGRA1:def_19;
then A87: indx (D2,D1,(len D1)) <= len D2 by FINSEQ_3:25;
then ((indx (D2,D1,j1)) + 1) + 1 <= len D2 by A83, XXREAL_0:2;
then A88: ((indx (D2,D1,j1)) + 1) + 1 in dom D2 by A85, FINSEQ_3:25;
then A89: D2 . (((indx (D2,D1,j1)) + 1) + 1) < D2 . (indx (D2,D1,(len D1))) by A83, A86, SEQM_3:def_1;
A90: 1 <= (indx (D2,D1,j1)) + 1 by A14, NAT_1:13;
A91: D1 . j1 = D2 . (indx (D2,D1,j1)) by A3, A8, INTEGRA1:def_19;
(indx (D2,D1,j1)) + 1 <= indx (D2,D1,(len D1)) by A83, A84, XXREAL_0:2;
then (indx (D2,D1,j1)) + 1 <= len D2 by A87, XXREAL_0:2;
then A92: (indx (D2,D1,j1)) + 1 in dom D2 by A90, FINSEQ_3:25;
A93: D1 . (len D1) = D2 . (indx (D2,D1,(len D1))) by A3, A5, INTEGRA1:def_19;
indx (D2,D1,j1) < (indx (D2,D1,j1)) + 1 by NAT_1:13;
then A94: D2 . (indx (D2,D1,j1)) < D2 . ((indx (D2,D1,j1)) + 1) by A13, A92, SEQM_3:def_1;
A95: D2 . ((indx (D2,D1,j1)) + 1) < D2 . (((indx (D2,D1,j1)) + 1) + 1) by A84, A92, A88, SEQM_3:def_1;
A96: ( not D2 . ((indx (D2,D1,j1)) + 1) in rng D1 & not D2 . (((indx (D2,D1,j1)) + 1) + 1) in rng D1 )
proof
assume A97: ( D2 . ((indx (D2,D1,j1)) + 1) in rng D1 or D2 . (((indx (D2,D1,j1)) + 1) + 1) in rng D1 ) ; ::_thesis: contradiction
percases ( D2 . ((indx (D2,D1,j1)) + 1) in rng D1 or D2 . (((indx (D2,D1,j1)) + 1) + 1) in rng D1 ) by A97;
suppose D2 . ((indx (D2,D1,j1)) + 1) in rng D1 ; ::_thesis: contradiction
then consider n being Element of NAT such that
A98: n in dom D1 and
A99: D1 . n = D2 . ((indx (D2,D1,j1)) + 1) by PARTFUN1:3;
j1 < n by A80, A94, A91, A98, A99, SEQ_4:137;
then A100: len D1 < n + 1 by XREAL_1:19;
D2 . ((indx (D2,D1,j1)) + 1) < D2 . (indx (D2,D1,(len D1))) by A95, A89, XXREAL_0:2;
then n < len D1 by A5, A93, A98, A99, SEQ_4:137;
hence contradiction by A100, NAT_1:13; ::_thesis: verum
end;
suppose D2 . (((indx (D2,D1,j1)) + 1) + 1) in rng D1 ; ::_thesis: contradiction
then consider n being Element of NAT such that
A101: n in dom D1 and
A102: D1 . n = D2 . (((indx (D2,D1,j1)) + 1) + 1) by PARTFUN1:3;
D2 . (indx (D2,D1,j1)) < D2 . (((indx (D2,D1,j1)) + 1) + 1) by A94, A95, XXREAL_0:2;
then j1 < n by A8, A91, A101, A102, SEQ_4:137;
then A103: len D1 < n + 1 by XREAL_1:19;
n < len D1 by A5, A89, A93, A101, A102, SEQ_4:137;
hence contradiction by A103, NAT_1:13; ::_thesis: verum
end;
end;
end;
D2 . ((indx (D2,D1,j1)) + 1) in rng D2 by A92, FUNCT_1:def_3;
then D2 . ((indx (D2,D1,j1)) + 1) in {x} by A4, A96, XBOOLE_0:def_3;
then A104: D2 . ((indx (D2,D1,j1)) + 1) = x by TARSKI:def_1;
D2 . (((indx (D2,D1,j1)) + 1) + 1) in rng D2 by A88, FUNCT_1:def_3;
then D2 . (((indx (D2,D1,j1)) + 1) + 1) in {x} by A4, A96, XBOOLE_0:def_3;
then D2 . ((indx (D2,D1,j1)) + 1) = D2 . (((indx (D2,D1,j1)) + 1) + 1) by A104, TARSKI:def_1;
hence contradiction by A84, A92, A88, SEQ_4:138; ::_thesis: verum
end;
A105: len D1 <= len (lower_volume (g,D1)) by INTEGRA1:def_7;
A106: 1 <= len D1 by A69, FINSEQ_1:1;
then A107: (mid ((lower_volume (g,D1)),(len D1),(len D1))) . 1 = (lower_volume (g,D1)) . (len D1) by A105, FINSEQ_6:118;
((len D1) -' (len D1)) + 1 = 1 by Lm1;
then len (mid ((lower_volume (g,D1)),(len D1),(len D1))) = 1 by A106, A105, FINSEQ_6:118;
then A108: mid ((lower_volume (g,D1)),(len D1),(len D1)) = <*((lower_volume (g,D1)) . (len D1))*> by A107, FINSEQ_1:40;
A109: 1 <= (indx (D2,D1,j1)) + 1 by A14, NAT_1:13;
indx (D2,D1,(len D1)) in dom D2 by A3, A5, INTEGRA1:def_19;
then A110: indx (D2,D1,(len D1)) in Seg (len D2) by FINSEQ_1:def_3;
then A111: 1 <= indx (D2,D1,(len D1)) by FINSEQ_1:1;
indx (D2,D1,(len D1)) in Seg (len (lower_volume (g,D2))) by A110, INTEGRA1:def_7;
then A112: indx (D2,D1,(len D1)) <= 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,(len D1))) -' ((indx (D2,D1,j1)) + 1) = (indx (D2,D1,(len D1))) - ((indx (D2,D1,j1)) + 1) by A79, XREAL_1:233;
then ((indx (D2,D1,(len D1))) -' ((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,(len D1))))) <= 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,(len D1))) -' ((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,(len D1))))) by A79, A111, A112, A109, A113, FINSEQ_6:118;
now__::_thesis:_(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)
percases ( len (mid ((lower_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = 1 or len (mid ((lower_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = 2 ) by A117, A115, Lm2;
supposeA118: len (mid ((lower_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = 1 ; ::_thesis: (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)
upper_bound (divset (D1,(len D1))) = D1 . (len D1) by A5, A7, INTEGRA1:def_4;
then A119: upper_bound (divset (D1,(len D1))) = D2 . (indx (D2,D1,(len D1))) by A3, A5, INTEGRA1:def_19;
lower_bound (divset (D1,(len D1))) = D1 . j1 by A5, A7, INTEGRA1:def_4;
then lower_bound (divset (D1,(len D1))) = D2 . (indx (D2,D1,j1)) by A3, A8, INTEGRA1:def_19;
then A120: divset (D1,(len D1)) = [.(D2 . (indx (D2,D1,j1))),(D2 . (indx (D2,D1,(len D1)))).] 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,(len D1)) in dom D2 by A3, A5, INTEGRA1:def_19;
len (mid ((lower_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = ((indx (D2,D1,(len D1))) -' ((indx (D2,D1,j1)) + 1)) + 1 by A79, A111, A112, A109, A113, FINSEQ_6:118;
then A124: (indx (D2,D1,(len D1))) - ((indx (D2,D1,j1)) + 1) = 0 by A79, A118, XREAL_1:233;
then indx (D2,D1,(len D1)) <> 1 by A13, FINSEQ_3:25;
then A125: upper_bound (divset (D2,(indx (D2,D1,(len D1))))) = D2 . (indx (D2,D1,(len D1))) by A123, INTEGRA1:def_4;
lower_bound (divset (D2,(indx (D2,D1,(len D1))))) = D2 . ((indx (D2,D1,(len D1))) - 1) by A14, A124, A123, INTEGRA1:def_4;
then A126: divset (D2,(indx (D2,D1,(len D1)))) = divset (D1,(len D1)) by A124, A120, A125, INTEGRA1:4;
(mid ((lower_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) . 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,(len D1)))) = <*((lower_volume (g,D2)) . ((indx (D2,D1,j1)) + 1))*> by A118, FINSEQ_1:40;
then Sum (mid ((lower_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = (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)) . (len D1) by A5, A124, A126, INTEGRA1:def_7
.= Sum (mid ((lower_volume (g,D1)),(len D1),(len D1))) by A108, FINSOP_1:11 ;
hence (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) by A121, A122; ::_thesis: verum
end;
supposeA127: len (mid ((lower_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = 2 ; ::_thesis: (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)
A128: (mid ((lower_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) . 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,(len D1))))) . 2 = H1(D2) . ((2 + ((indx (D2,D1,j1)) + 1)) -' 1) by A79, A111, A112, A109, A113, A127, FINSEQ_6:118
.= H1(D2) . ((2 + ((indx (D2,D1,j1)) + 1)) - 1) by A129, XREAL_1:233
.= H1(D2) . ((indx (D2,D1,j1)) + (1 + 1)) ;
then mid ((lower_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))) = <*((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,(len D1))))) = ((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,(len D1))) = D1 . (len D1) by A5, A7, INTEGRA1:def_4;
then A132: upper_bound (divset (D1,(len D1))) = D2 . (indx (D2,D1,(len D1))) by A3, A5, INTEGRA1:def_19;
A133: vol (divset (D2,((indx (D2,D1,j1)) + 2))) >= 0 by INTEGRA1:9;
((indx (D2,D1,(len D1))) -' ((indx (D2,D1,j1)) + 1)) + 1 = 2 by A79, A111, A112, A109, A113, A127, FINSEQ_6:118;
then A134: ((indx (D2,D1,(len D1))) - ((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,(len D1))) = D1 . j1 by A5, A7, INTEGRA1:def_4;
then lower_bound (divset (D1,(len D1))) = D2 . (indx (D2,D1,j1)) by A3, A8, INTEGRA1:def_19;
then A136: vol (divset (D1,(len D1))) = (((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,(len D1))) = ((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,(len D1))) = (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)) . (len D1) = (lower_bound (rng (g | (divset (D1,(len D1)))))) * ((vol (divset (D2,((indx (D2,D1,j1)) + 1)))) + (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) by A5, INTEGRA1:def_7;
A145: (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) - (Sum (mid (H1(D1),(len D1),(len D1)))) <= ((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;
set ID2 = (indx (D2,D1,j1)) + 2;
set IR = (lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))));
divset (D1,(len D1)) c= A by A5, INTEGRA1:8;
then A146: lower_bound (rng (g | (divset (D1,(len D1))))) >= lower_bound (rng g) by A72, Lm4;
Sum (mid (H1(D1),(len D1),(len D1))) = ((lower_bound (rng (g | (divset (D1,(len D1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((lower_bound (rng (g | (divset (D1,(len D1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) by A108, A144, FINSOP_1:11;
then (Sum (mid (H1(D1),(len D1),(len D1)))) - ((lower_bound (rng (g | (divset (D1,(len D1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) >= (lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) by A133, A146, XREAL_1:64;
then Sum (mid (H1(D1),(len D1),(len D1))) >= ((lower_bound (rng (g | (divset (D1,(len D1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) + ((lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) by XREAL_1:19;
then A147: (Sum (mid (H1(D1),(len D1),(len D1)))) - ((lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) >= (lower_bound (rng (g | (divset (D1,(len D1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by XREAL_1:19;
(lower_bound (rng (g | (divset (D1,(len D1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) >= (lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A131, A146, XREAL_1:64;
then (Sum (mid (H1(D1),(len D1),(len D1)))) - ((lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) >= (lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A147, XXREAL_0:2;
then A148: Sum (mid (H1(D1),(len D1),(len D1))) >= ((lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) by XREAL_1:19;
((indx (D2,D1,(len D1))) -' ((indx (D2,D1,j1)) + 1)) + 1 = 2 by A79, A111, A112, A109, A113, A127, FINSEQ_6:118;
then A149: ((indx (D2,D1,(len D1))) - ((indx (D2,D1,j1)) + 1)) + 1 = 2 by A79, XREAL_1:233;
(indx (D2,D1,j1)) + 1 in dom D2 by A114, FINSEQ_1:def_3;
then divset (D2,((indx (D2,D1,j1)) + 1)) c= A by INTEGRA1:8;
then lower_bound (rng (g | (divset (D2,((indx (D2,D1,j1)) + 1))))) <= upper_bound (rng g) by A72, Lm4;
then A150: (lower_bound (rng (g | (divset (D2,((indx (D2,D1,j1)) + 1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) <= (upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A131, XREAL_1:64;
A151: indx (D2,D1,(len D1)) in dom D2 by A3, A5, INTEGRA1:def_19;
then divset (D2,((indx (D2,D1,j1)) + 2)) c= A by A149, INTEGRA1:8;
then A152: lower_bound (rng (g | (divset (D2,((indx (D2,D1,j1)) + 2))))) <= upper_bound (rng g) by A72, Lm4;
Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = ((lower_bound (rng (g | (divset (D2,((indx (D2,D1,j1)) + 2)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + (H1(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 (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) - ((lower_bound (rng (g | (divset (D2,((indx (D2,D1,j1)) + 1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) <= (upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) by A133, A152, XREAL_1:64;
then Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) <= ((upper_bound (rng g)) * (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 XREAL_1:20;
then (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) - ((upper_bound (rng g)) * (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 XREAL_1:20;
then (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) - ((upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) <= (upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A150, XXREAL_0:2;
then Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) <= ((upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) by XREAL_1:20;
then (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) - (Sum (mid (H1(D1),(len D1),(len D1)))) <= (((upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))))) - (((lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))))) by A148, XREAL_1:13;
hence (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) - (Sum (mid (H1(D1),(len D1),(len D1)))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * ((vol (divset (D2,((indx (D2,D1,j1)) + 2)))) + (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) ; ::_thesis: verum
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,(len D1)))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1) by A5, Lm5, XREAL_1:64;
hence (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) by A143, A145, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (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) ; ::_thesis: verum
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,(len D1))) = D1 . (len D1) by A3, A5, INTEGRA1:def_19;
A156: indx (D2,D1,(len D1)) >= len (lower_volume (g,D2))
proof
assume indx (D2,D1,(len D1)) < len (lower_volume (g,D2)) ; ::_thesis: contradiction
then indx (D2,D1,(len D1)) < 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 ; ::_thesis: contradiction
D2 . (len D2) <= upper_bound A by INTEGRA1:def_2;
hence contradiction by A157, INTEGRA1:def_2; ::_thesis: verum
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; ::_thesis: verum
end;
indx (D2,D1,(len D1)) in Seg (len D2) by A6, FINSEQ_1:def_3;
then indx (D2,D1,(len D1)) in Seg (len (lower_volume (g,D2))) by INTEGRA1:def_7;
then indx (D2,D1,(len D1)) in dom (lower_volume (g,D2)) by FINSEQ_1:def_3;
then A159: H2(D2, indx (D2,D1,(len D1))) = Sum ((lower_volume (g,D2)) | (indx (D2,D1,(len D1)))) by INTEGRA1:def_20
.= Sum (lower_volume (g,D2)) by A156, FINSEQ_1:58 ;
len D1 in Seg (len (lower_volume (g,D1))) by A69, INTEGRA1:def_7;
then len D1 in dom (lower_volume (g,D1)) by FINSEQ_1:def_3;
then A160: H2(D1, len D1) = Sum ((lower_volume (g,D1)) | (len D1)) 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: H2(D2, indx (D2,D1,j1)) = Sum ((lower_volume (g,D2)) | (indx (D2,D1,j1))) by INTEGRA1:def_20
.= H2(D1,j1) by A162, A161, INTEGRA1:def_20 ;
indx (D2,D1,(len D1)) <= len D2 by A76, FINSEQ_1:1;
then A164: indx (D2,D1,(len D1)) <= len H1(D2) by INTEGRA1:def_7;
A165: len D2 = len H1(D2) by INTEGRA1:def_7;
then A166: indx (D2,D1,(len D1)) in dom H1(D2) by A75, FINSEQ_3:29;
indx (D2,D1,j1) in dom H1(D2) by A13, A165, FINSEQ_3:29;
then H2(D2, indx (D2,D1,j1)) = Sum (H1(D2) | (indx (D2,D1,j1))) by INTEGRA1:def_20;
then H2(D2, indx (D2,D1,j1)) + (Sum (mid ((lower_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) = Sum ((H1(D2) | (indx (D2,D1,j1))) ^ (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) by RVSUM_1:75
.= Sum ((mid (H1(D2),1,(indx (D2,D1,j1)))) ^ (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) by A14, FINSEQ_6:116
.= Sum (mid (H1(D2),1,(indx (D2,D1,(len D1))))) by A14, A78, A164, INTEGRA2:4
.= Sum (H1(D2) | (indx (D2,D1,(len D1)))) by A77, FINSEQ_6:116 ;
then H2(D2, indx (D2,D1,j1)) + (Sum (mid ((lower_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) = H2(D2, indx (D2,D1,(len D1))) by A166, INTEGRA1:def_20;
hence (Sum (lower_volume (g,D2))) - (Sum (lower_volume (g,D1))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1) by A163, A81, A74, A159, A160; ::_thesis: verum
end;
theorem Th11: :: INTEGRA3:11
for x being Real
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for g being Function of A,REAL st x in divset (D1,(len D1)) & len D1 >= 2 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & g | A is bounded holds
(Sum (upper_volume (g,D1))) - (Sum (upper_volume (g,D2))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1)
proof
let x be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for g being Function of A,REAL st x in divset (D1,(len D1)) & len D1 >= 2 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & g | A is bounded holds
(Sum (upper_volume (g,D1))) - (Sum (upper_volume (g,D2))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1)
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for g being Function of A,REAL st x in divset (D1,(len D1)) & len D1 >= 2 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & g | A is bounded holds
(Sum (upper_volume (g,D1))) - (Sum (upper_volume (g,D2))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1)
let D1, D2 be Division of A; ::_thesis: for g being Function of A,REAL st x in divset (D1,(len D1)) & len D1 >= 2 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & g | A is bounded holds
(Sum (upper_volume (g,D1))) - (Sum (upper_volume (g,D2))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1)
let g be Function of A,REAL; ::_thesis: ( 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) )
assume that
A1: x in divset (D1,(len D1)) and
A2: len D1 >= 2 ; ::_thesis: ( not D1 <= D2 or not rng D2 = (rng D1) \/ {x} or not g | A is bounded or (Sum (upper_volume (g,D1))) - (Sum (upper_volume (g,D2))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1) )
set j = len D1;
assume that
A3: D1 <= D2 and
A4: rng D2 = (rng D1) \/ {x} ; ::_thesis: ( not g | A is bounded or (Sum (upper_volume (g,D1))) - (Sum (upper_volume (g,D2))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1) )
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,(len D1)) in dom D2 by A3, INTEGRA1:def_19;
deffunc H1( Division of A) -> FinSequence of REAL = upper_volume (g,$1);
deffunc H2( Division of A, Nat) -> Element of REAL = (PartSums (upper_volume (g,$1))) . $2;
A8: len D1 >= 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,(len D1))) = D1 . (len D1) by A3, A6, INTEGRA1:def_19;
A18: indx (D2,D1,(len D1)) >= len (upper_volume (g,D2))
proof
assume indx (D2,D1,(len D1)) < len (upper_volume (g,D2)) ; ::_thesis: contradiction
then indx (D2,D1,(len D1)) < 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 ; ::_thesis: contradiction
D2 . (len D2) <= upper_bound A by INTEGRA1:def_2;
hence contradiction by A19, INTEGRA1:def_2; ::_thesis: verum
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; ::_thesis: verum
end;
indx (D2,D1,(len D1)) in Seg (len D2) by A7, FINSEQ_1:def_3;
then indx (D2,D1,(len D1)) in Seg (len (upper_volume (g,D2))) by INTEGRA1:def_6;
then indx (D2,D1,(len D1)) in dom (upper_volume (g,D2)) by FINSEQ_1:def_3;
then A21: H2(D2, indx (D2,D1,(len D1))) = Sum ((upper_volume (g,D2)) | (indx (D2,D1,(len D1)))) by INTEGRA1:def_20
.= Sum (upper_volume (g,D2)) by A18, FINSEQ_1:58 ;
indx (D2,D1,(len D1)) in dom D2 by A3, A6, INTEGRA1:def_19;
then A22: indx (D2,D1,(len D1)) in Seg (len D2) by FINSEQ_1:def_3;
then A23: 1 <= indx (D2,D1,(len D1)) 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 ; ::_thesis: (Sum (upper_volume (g,D1))) - (Sum (upper_volume (g,D2))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1)
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,(len D1))) - (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,(len D1))) - (indx (D2,D1,j1)) > 2 ; ::_thesis: contradiction
then A30: (indx (D2,D1,j1)) + (1 + 1) < indx (D2,D1,(len D1)) 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,(len D1)) in dom D2 by A3, A6, INTEGRA1:def_19;
then A34: indx (D2,D1,(len D1)) <= 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,(len D1))) 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,(len D1)) 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 . (len D1) = D2 . (indx (D2,D1,(len D1))) 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 ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( D2 . ID1 in rng D1 or D2 . ID2 in rng D1 ) by A44;
suppose D2 . ID1 in rng D1 ; ::_thesis: contradiction
then consider n being Element of NAT 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: len D1 < n + 1 by XREAL_1:19;
D2 . ID1 < D2 . (indx (D2,D1,(len D1))) by A42, A36, XXREAL_0:2;
then n < len D1 by A6, A40, A45, A46, SEQ_4:137;
hence contradiction by A47, NAT_1:13; ::_thesis: verum
end;
suppose D2 . ID2 in rng D1 ; ::_thesis: contradiction
then consider n being Element of NAT 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: len D1 < n + 1 by XREAL_1:19;
n < len D1 by A6, A36, A40, A48, A49, SEQ_4:137;
hence contradiction by A50, NAT_1:13; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
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; ::_thesis: verum
end;
A52: len D1 <= len (upper_volume (g,D1)) by INTEGRA1:def_6;
A53: 1 <= len D1 by A5, FINSEQ_1:1;
then A54: (mid ((upper_volume (g,D1)),(len D1),(len D1))) . 1 = (upper_volume (g,D1)) . (len D1) by A52, FINSEQ_6:118;
((len D1) -' (len D1)) + 1 = 1 by Lm1;
then len (mid ((upper_volume (g,D1)),(len D1),(len D1))) = 1 by A53, A52, FINSEQ_6:118;
then mid ((upper_volume (g,D1)),(len D1),(len D1)) = <*((upper_volume (g,D1)) . (len D1))*> by A54, FINSEQ_1:40;
then A55: Sum (mid ((upper_volume (g,D1)),(len D1),(len D1))) = (upper_volume (g,D1)) . (len D1) by FINSOP_1:11;
A56: 1 <= (indx (D2,D1,j1)) + 1 by A12, NAT_1:13;
indx (D2,D1,(len D1)) in dom D2 by A3, A6, INTEGRA1:def_19;
then A57: indx (D2,D1,(len D1)) in Seg (len D2) by FINSEQ_1:def_3;
then A58: 1 <= indx (D2,D1,(len D1)) by FINSEQ_1:1;
indx (D2,D1,(len D1)) in Seg (len (upper_volume (g,D2))) by A57, INTEGRA1:def_6;
then A59: indx (D2,D1,(len D1)) <= 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,(len D1))) -' ((indx (D2,D1,j1)) + 1) = (indx (D2,D1,(len D1))) - ((indx (D2,D1,j1)) + 1) by A15, XREAL_1:233;
then ((indx (D2,D1,(len D1))) -' ((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,(len D1))))) <= 2 by A15, A58, A59, A56, A60, FINSEQ_6:118;
((indx (D2,D1,(len D1))) -' ((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,(len D1))))) by A15, A58, A59, A56, A60, FINSEQ_6:118;
now__::_thesis:_(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)
percases ( len (mid ((upper_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = 1 or len (mid ((upper_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = 2 ) by A64, A63, Lm2;
supposeA65: len (mid ((upper_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = 1 ; ::_thesis: (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)
upper_bound (divset (D1,(len D1))) = D1 . (len D1) by A6, A9, INTEGRA1:def_4;
then A66: upper_bound (divset (D1,(len D1))) = D2 . (indx (D2,D1,(len D1))) by A3, A6, INTEGRA1:def_19;
lower_bound (divset (D1,(len D1))) = D1 . j1 by A6, A9, INTEGRA1:def_4;
then lower_bound (divset (D1,(len D1))) = D2 . (indx (D2,D1,j1)) by A3, A10, INTEGRA1:def_19;
then A67: divset (D1,(len D1)) = [.(D2 . (indx (D2,D1,j1))),(D2 . (indx (D2,D1,(len D1)))).] 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,(len D1)) in dom D2 by A3, A6, INTEGRA1:def_19;
len (mid ((upper_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = ((indx (D2,D1,(len D1))) -' ((indx (D2,D1,j1)) + 1)) + 1 by A15, A58, A59, A56, A60, FINSEQ_6:118;
then A71: (indx (D2,D1,(len D1))) - ((indx (D2,D1,j1)) + 1) = 0 by A15, A65, XREAL_1:233;
then indx (D2,D1,(len D1)) <> 1 by A11, FINSEQ_3:25;
then A72: upper_bound (divset (D2,(indx (D2,D1,(len D1))))) = D2 . (indx (D2,D1,(len D1))) by A70, INTEGRA1:def_4;
lower_bound (divset (D2,(indx (D2,D1,(len D1))))) = D2 . ((indx (D2,D1,(len D1))) - 1) by A12, A71, A70, INTEGRA1:def_4;
then A73: divset (D2,(indx (D2,D1,(len D1)))) = divset (D1,(len D1)) by A71, A67, A72, INTEGRA1:4;
(mid ((upper_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) . 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,(len D1)))) = <*((upper_volume (g,D2)) . ((indx (D2,D1,j1)) + 1))*> by A65, FINSEQ_1:40;
then Sum (mid ((upper_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = (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)),(len D1),(len D1))) by A6, A55, A71, A73, INTEGRA1:def_6 ;
hence (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) by A68, A69; ::_thesis: verum
end;
supposeA74: len (mid ((upper_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = 2 ; ::_thesis: (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)
A75: (mid ((upper_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) . 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,(len D1))))) . 2 = H1(D2) . ((2 + ((indx (D2,D1,j1)) + 1)) -' 1) by A15, A58, A59, A56, A60, A74, FINSEQ_6:118
.= H1(D2) . ((2 + ((indx (D2,D1,j1)) + 1)) - 1) by A76, XREAL_1:233
.= H1(D2) . ((indx (D2,D1,j1)) + (1 + 1)) ;
then mid ((upper_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))) = <*((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,(len D1))))) = ((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,(len D1))) = D1 . (len D1) by A6, A9, INTEGRA1:def_4;
then A79: upper_bound (divset (D1,(len D1))) = D2 . (indx (D2,D1,(len D1))) by A3, A6, INTEGRA1:def_19;
A80: vol (divset (D2,((indx (D2,D1,j1)) + 2))) >= 0 by INTEGRA1:9;
((indx (D2,D1,(len D1))) -' ((indx (D2,D1,j1)) + 1)) + 1 = 2 by A15, A58, A59, A56, A60, A74, FINSEQ_6:118;
then A81: ((indx (D2,D1,(len D1))) - ((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,(len D1))) = D1 . j1 by A6, A9, INTEGRA1:def_4;
then lower_bound (divset (D1,(len D1))) = D2 . (indx (D2,D1,j1)) by A3, A10, INTEGRA1:def_19;
then A83: vol (divset (D1,(len D1))) = (((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,(len D1))) = ((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,(len D1))) = (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)) . (len D1) = (upper_bound (rng (g | (divset (D1,(len D1)))))) * ((vol (divset (D2,((indx (D2,D1,j1)) + 1)))) + (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) by A6, INTEGRA1:def_6;
A92: (Sum (mid (H1(D1),(len D1),(len D1)))) - (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) <= ((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;
set ID2 = (indx (D2,D1,j1)) + 2;
A93: (Sum (mid (H1(D1),(len D1),(len D1)))) - ((upper_bound (rng (g | (divset (D1,(len D1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) = (upper_bound (rng (g | (divset (D1,(len D1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) by A55, A91;
divset (D1,(len D1)) c= A by A6, INTEGRA1:8;
then A94: upper_bound (rng (g | (divset (D1,(len D1))))) <= upper_bound (rng g) by A27, Lm4;
then (upper_bound (rng (g | (divset (D1,(len D1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) <= (upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) by A80, XREAL_1:64;
then Sum (mid (H1(D1),(len D1),(len D1))) <= ((upper_bound (rng (g | (divset (D1,(len D1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) + ((upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) by A93, XREAL_1:20;
then A95: (Sum (mid (H1(D1),(len D1),(len D1)))) - ((upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) <= (upper_bound (rng (g | (divset (D1,(len D1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by XREAL_1:20;
(upper_bound (rng (g | (divset (D1,(len D1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) <= (upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A78, A94, XREAL_1:64;
then (Sum (mid (H1(D1),(len D1),(len D1)))) - ((upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) <= (upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A95, XXREAL_0:2;
then A96: Sum (mid (H1(D1),(len D1),(len D1))) <= ((upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) by XREAL_1:20;
(indx (D2,D1,j1)) + 1 in dom D2 by A61, FINSEQ_1:def_3;
then divset (D2,((indx (D2,D1,j1)) + 1)) c= A by INTEGRA1:8;
then upper_bound (rng (g | (divset (D2,((indx (D2,D1,j1)) + 1))))) >= lower_bound (rng g) by A27, Lm4;
then A97: (upper_bound (rng (g | (divset (D2,((indx (D2,D1,j1)) + 1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) >= (lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A78, XREAL_1:64;
((indx (D2,D1,(len D1))) -' ((indx (D2,D1,j1)) + 1)) + 1 = 2 by A15, A58, A59, A56, A60, A74, FINSEQ_6:118;
then A98: ((indx (D2,D1,(len D1))) - ((indx (D2,D1,j1)) + 1)) + 1 = 2 by A15, XREAL_1:233;
A99: indx (D2,D1,(len D1)) in dom D2 by A3, A6, INTEGRA1:def_19;
then divset (D2,((indx (D2,D1,j1)) + 2)) c= A by A98, INTEGRA1:8;
then A100: upper_bound (rng (g | (divset (D2,((indx (D2,D1,j1)) + 2))))) >= lower_bound (rng g) by A27, Lm4;
Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) = ((upper_bound (rng (g | (divset (D2,((indx (D2,D1,j1)) + 2)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + (H1(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 (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) - ((upper_bound (rng (g | (divset (D2,((indx (D2,D1,j1)) + 1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) >= (lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) by A80, A100, XREAL_1:64;
then Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) >= ((lower_bound (rng g)) * (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 XREAL_1:19;
then (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) - ((lower_bound (rng g)) * (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 XREAL_1:19;
then (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) - ((lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) >= (lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A97, XXREAL_0:2;
then Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1))))) >= ((lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) by XREAL_1:19;
then (Sum (mid (H1(D1),(len D1),(len D1)))) - (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) <= (((upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((upper_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))))) - (((lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((lower_bound (rng g)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))))) by A96, XREAL_1:13;
hence (Sum (mid (H1(D1),(len D1),(len D1)))) - (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * ((vol (divset (D2,((indx (D2,D1,j1)) + 2)))) + (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) ; ::_thesis: verum
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,(len D1)))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1) by A6, Lm5, XREAL_1:64;
hence (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) by A90, A92, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (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) ; ::_thesis: verum
end;
len D1 in Seg (len (upper_volume (g,D1))) by A5, INTEGRA1:def_6;
then len D1 in dom (upper_volume (g,D1)) by FINSEQ_1:def_3;
then A101: H2(D1, len D1) = Sum ((upper_volume (g,D1)) | (len D1)) by INTEGRA1:def_20
.= Sum (upper_volume (g,D1)) by A8, FINSEQ_1:58 ;
A102: len D1 <= len H1(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, INTEGRA1:35;
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 being Element of NAT st 1 <= k & k <= j1 holds
k = indx (D2,D1,k)
proof
let k be Element of NAT ; ::_thesis: ( 1 <= k & k <= j1 implies k = indx (D2,D1,k) )
assume that
A108: 1 <= k and
A109: k <= j1 ; ::_thesis: k = indx (D2,D1,k)
assume A110: k <> indx (D2,D1,k) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( k > indx (D2,D1,k) or k < indx (D2,D1,k) ) by A110, XXREAL_0:1;
supposeA111: k > indx (D2,D1,k) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
supposeA118: k < indx (D2,D1,k) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A125: for k being 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; ::_thesis: ( 1 <= k & k <= len ((upper_volume (g,D1)) | j1) implies ((upper_volume (g,D1)) | j1) . k = ((upper_volume (g,D2)) | (indx (D2,D1,j1))) . k )
assume that
A127: 1 <= k and
A128: k <= len ((upper_volume (g,D1)) | j1) ; ::_thesis: ((upper_volume (g,D1)) | j1) . k = ((upper_volume (g,D2)) | (indx (D2,D1,j1))) . k
reconsider k = 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
percases ( k = 1 or k <> 1 ) ;
supposeA139: k = 1 ; ::_thesis: ( 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)))) )
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 ( 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)))) ) by A133, A137, A141, A140, INTEGRA1:def_4; ::_thesis: verum
end;
supposeA142: k <> 1 ; ::_thesis: ( 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)))) )
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 ( 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)))) ) by A3, A132, A149, A144, A147, A148, A145, INTEGRA1:def_19; ::_thesis: verum
end;
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 ((upper_volume (g,D1)) | j1) . k = ((upper_volume (g,D2)) | (indx (D2,D1,j1))) . k by A152; ::_thesis: verum
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;
len D1 < (len D1) + 1 by NAT_1:13;
then A158: j1 < len D1 by XREAL_1:19;
indx (D2,D1,(len D1)) <= len D2 by A22, FINSEQ_1:1;
then A159: indx (D2,D1,(len D1)) <= len H1(D2) by INTEGRA1:def_6;
then A160: indx (D2,D1,(len D1)) in dom H1(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 H1(D2)) by INTEGRA1:def_6;
then indx (D2,D1,j1) in dom H1(D2) by FINSEQ_1:def_3;
then H2(D2, indx (D2,D1,j1)) = Sum (H1(D2) | (indx (D2,D1,j1))) by INTEGRA1:def_20;
then H2(D2, indx (D2,D1,j1)) + (Sum (mid ((upper_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) = Sum ((H1(D2) | (indx (D2,D1,j1))) ^ (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) by RVSUM_1:75
.= Sum ((mid (H1(D2),1,(indx (D2,D1,j1)))) ^ (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) by A12, FINSEQ_6:116
.= Sum (mid (H1(D2),1,(indx (D2,D1,(len D1))))) by A12, A14, A159, INTEGRA2:4
.= Sum (H1(D2) | (indx (D2,D1,(len D1)))) by A23, FINSEQ_6:116 ;
then A161: H2(D2, indx (D2,D1,j1)) + (Sum (mid ((upper_volume (g,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,(len D1)))))) = H2(D2, indx (D2,D1,(len D1))) by A160, INTEGRA1:def_20;
A162: 1 <= len D1 by A5, FINSEQ_1:1;
then A163: len D1 in dom H1(D1) by A102, FINSEQ_3:25;
j1 in Seg (len D1) by A10, FINSEQ_1:def_3;
then j1 in Seg (len H1(D1)) by INTEGRA1:def_6;
then j1 in dom H1(D1) by FINSEQ_1:def_3;
then H2(D1,j1) = Sum (H1(D1) | j1) by INTEGRA1:def_20;
then H2(D1,j1) + (Sum (mid (H1(D1),(len D1),(len D1)))) = Sum ((H1(D1) | j1) ^ (mid (H1(D1),(len D1),(len D1)))) by RVSUM_1:75
.= Sum ((mid (H1(D1),1,j1)) ^ (mid (H1(D1),(j1 + 1),(len D1)))) by A103, FINSEQ_6:116
.= Sum (mid (H1(D1),1,(len D1))) by A103, A102, A158, INTEGRA2:4
.= Sum (H1(D1) | (len D1)) by A162, FINSEQ_6:116 ;
then A164: H2(D1,j1) + (Sum (mid ((upper_volume (g,D1)),(len D1),(len D1)))) = H2(D1, len D1) 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 H2(D2, indx (D2,D1,j1)) = Sum ((upper_volume (g,D2)) | (indx (D2,D1,j1))) by INTEGRA1:def_20
.= H2(D1,j1) by A156, A157, INTEGRA1:def_20 ;
hence (Sum (upper_volume (g,D1))) - (Sum (upper_volume (g,D2))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1) by A28, A161, A164, A21, A101; ::_thesis: verum
end;
Lm7: for y being Real
for A being non empty closed_interval Subset of REAL
for f being Function of A,REAL st vol A <> 0 & y in rng (lower_sum_set f) holds
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
let y be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for f being Function of A,REAL st vol A <> 0 & y in rng (lower_sum_set f) holds
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 )
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being Function of A,REAL st vol A <> 0 & y in rng (lower_sum_set f) holds
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 )
let f be Function of A,REAL; ::_thesis: ( 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 ) )
assume A1: vol A <> 0 ; ::_thesis: ( not y in rng (lower_sum_set f) or 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 ) )
assume y in rng (lower_sum_set f) ; ::_thesis: 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 )
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 = 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__::_thesis:_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_)
percases ( D3 . 1 <> lower_bound A or D3 . 1 = lower_bound A ) ;
supposeA6: D3 . 1 <> lower_bound A ; ::_thesis: 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 )
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 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 ) by A2, A3; ::_thesis: verum
end;
supposeA7: D3 . 1 = lower_bound A ; ::_thesis: 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 )
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 = 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, XBOOLE_1:1;
then reconsider D = 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;
then A22: len ((lower_volume (f,D3)) | 1) = 1 by FINSEQ_1:59;
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 being 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; ::_thesis: ( 1 <= i & i <= len (mid ((lower_volume (f,D3)),2,(len (lower_volume (f,D3))))) implies (mid ((lower_volume (f,D3)),2,(len (lower_volume (f,D3))))) . i = (lower_volume (f,D)) . i )
assume that
A25: 1 <= i and
A26: i <= len (mid ((lower_volume (f,D3)),2,(len (lower_volume (f,D3))))) ; ::_thesis: (mid ((lower_volume (f,D3)),2,(len (lower_volume (f,D3))))) . i = (lower_volume (f,D)) . i
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;
percases ( i = 1 or i <> 1 ) ;
supposeA37: i = 1 ; ::_thesis: divset (D3,(i + 1)) = divset (D,i)
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 divset (D3,(i + 1)) = divset (D,i) by A39, A38, INTEGRA1:4; ::_thesis: verum
end;
supposeA40: i <> 1 ; ::_thesis: divset (D3,(i + 1)) = divset (D,i)
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 divset (D3,(i + 1)) = divset (D,i) by INTEGRA1:4; ::_thesis: verum
end;
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 ;
i in NAT by ORDINAL1:def_12;
then (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 (mid ((lower_volume (f,D3)),2,(len (lower_volume (f,D3))))) . i = (lower_volume (f,D)) . i by A44, A30, INTEGRA1:def_7; ::_thesis: verum
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 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 ) by A7, A48, A47, A18; ::_thesis: verum
end;
hence 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 ) ; ::_thesis: verum
end;
end;
end;
hence 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 ) ; ::_thesis: verum
end;
theorem Th12: :: INTEGRA3:12
for r being Real
for i, j being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D & j in dom D & i <= j & r < (mid (D,i,j)) . 1 holds
ex B being 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
let r be Real; ::_thesis: for i, j being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D & j in dom D & i <= j & r < (mid (D,i,j)) . 1 holds
ex B being 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 )
let i, j be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D & j in dom D & i <= j & r < (mid (D,i,j)) . 1 holds
ex B being 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 )
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A st i in dom D & j in dom D & i <= j & r < (mid (D,i,j)) . 1 holds
ex B being 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 )
let D be Division of A; ::_thesis: ( i in dom D & j in dom D & i <= j & r < (mid (D,i,j)) . 1 implies ex B being 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 ) )
assume A1: i in dom D ; ::_thesis: ( not j in dom D or not i <= j or not r < (mid (D,i,j)) . 1 or ex B being 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 ) )
assume A2: j in dom D ; ::_thesis: ( not i <= j or not r < (mid (D,i,j)) . 1 or ex B being 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 ) )
assume i <= j ; ::_thesis: ( not r < (mid (D,i,j)) . 1 or ex B being 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 ) )
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 ; ::_thesis: ex B being 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 )
ex a, b being Real 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 = [.r,(upper_bound C).] 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 Real st x in C holds
x in B
proof
let x be Real; ::_thesis: ( x in C implies x in B )
assume A10: x in C ; ::_thesis: x in B
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 x in B by A8, A9, A11, INTEGRA2:1; ::_thesis: verum
end;
then A12: C c= B by SUBSET_1:2;
rng (mid (D,i,j)) c= C by A5, INTEGRA1:def_2;
then rng (mid (D,i,j)) c= B by A12, XBOOLE_1:1;
then MD is Division of B by A4, A9, INTEGRA1:def_2;
hence ex B being 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 ) by A4, A8, A9; ::_thesis: verum
end;
Lm8: for A being non empty closed_interval Subset of REAL
for D1 being Division of A st vol A <> 0 & len D1 = 1 holds
<*(lower_bound A)*> ^ D1 is non empty increasing FinSequence of REAL
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1 being Division of A st vol A <> 0 & len D1 = 1 holds
<*(lower_bound A)*> ^ D1 is non empty increasing FinSequence of REAL
let D1 be Division of A; ::_thesis: ( vol A <> 0 & len D1 = 1 implies <*(lower_bound A)*> ^ D1 is non empty increasing FinSequence of REAL )
assume A1: vol A <> 0 ; ::_thesis: ( not len D1 = 1 or <*(lower_bound A)*> ^ D1 is non empty increasing FinSequence of REAL )
set MD1 = <*(lower_bound A)*> ^ D1;
A2: vol A >= 0 by INTEGRA1:9;
assume len D1 = 1 ; ::_thesis: <*(lower_bound A)*> ^ D1 is non empty increasing FinSequence of REAL
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 Element of NAT st n in dom (<*(lower_bound A)*> ^ D1) & m in dom (<*(lower_bound A)*> ^ D1) & n < m holds
(<*(lower_bound A)*> ^ D1) . n < (<*(lower_bound A)*> ^ D1) . m
proof
let n, m be Element of NAT ; ::_thesis: ( n in dom (<*(lower_bound A)*> ^ D1) & m in dom (<*(lower_bound A)*> ^ D1) & n < m implies (<*(lower_bound A)*> ^ D1) . n < (<*(lower_bound A)*> ^ D1) . m )
assume that
A5: n in dom (<*(lower_bound A)*> ^ D1) and
A6: m in dom (<*(lower_bound A)*> ^ D1) and
A7: n < m ; ::_thesis: (<*(lower_bound A)*> ^ D1) . n < (<*(lower_bound A)*> ^ D1) . m
A8: not m in dom <*(lower_bound A)*>
proof
assume m in dom <*(lower_bound A)*> ; ::_thesis: contradiction
then m in Seg (len <*(lower_bound A)*>) by FINSEQ_1:def_3;
then m in {1} by FINSEQ_1:2, FINSEQ_1:39;
then A9: n < 1 by A7, TARSKI:def_1;
n in Seg (len (<*(lower_bound A)*> ^ D1)) by A5, FINSEQ_1:def_3;
hence contradiction by A9, FINSEQ_1:1; ::_thesis: verum
end;
A10: not (<*(lower_bound A)*> ^ D1) . m in rng <*(lower_bound A)*>
proof
assume (<*(lower_bound A)*> ^ D1) . m in rng <*(lower_bound A)*> ; ::_thesis: contradiction
then (<*(lower_bound A)*> ^ D1) . m in {(lower_bound A)} by FINSEQ_1:38;
then A11: (<*(lower_bound A)*> ^ D1) . 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 = (<*(lower_bound A)*> ^ D1) . m by A13, A14, FINSEQ_1:def_7;
hence contradiction by A3, A11, A13, A15, A12, SEQ_4:137, XREAL_1:47; ::_thesis: verum
end;
(<*(lower_bound A)*> ^ D1) . m in rng (<*(lower_bound A)*> ^ D1) by A6, FUNCT_1:def_3;
then (<*(lower_bound A)*> ^ D1) . m in (rng <*(lower_bound A)*>) \/ (rng D1) by FINSEQ_1:31;
then A16: (<*(lower_bound A)*> ^ D1) . m in rng D1 by A10, XBOOLE_0:def_3;
now__::_thesis:_(<*(lower_bound_A)*>_^_D1)_._n_<_(<*(lower_bound_A)*>_^_D1)_._m
percases ( n in dom <*(lower_bound A)*> or ex i being Nat st
( i in dom D1 & n = (len <*(lower_bound A)*>) + i ) ) by A5, FINSEQ_1:25;
supposeA17: n in dom <*(lower_bound A)*> ; ::_thesis: (<*(lower_bound A)*> ^ D1) . n < (<*(lower_bound A)*> ^ D1) . m
then n in Seg (len <*(lower_bound A)*>) by FINSEQ_1:def_3;
then n in {1} by FINSEQ_1:2, FINSEQ_1:39;
then A18: n = 1 by TARSKI:def_1;
A19: (<*(lower_bound A)*> ^ D1) . n = <*(lower_bound A)*> . n by A17, FINSEQ_1:def_7
.= lower_bound A by A18, FINSEQ_1:def_8 ;
rng D1 <> {} ;
then A20: 1 in dom D1 by FINSEQ_3:32;
consider k being Element of NAT such that
A21: k in dom D1 and
A22: (<*(lower_bound A)*> ^ D1) . m = D1 . k by A16, PARTFUN1:3;
1 <= k by A21, FINSEQ_3:25;
then D1 . 1 <= (<*(lower_bound A)*> ^ D1) . m by A21, A22, A20, SEQ_4:137;
hence (<*(lower_bound A)*> ^ D1) . n < (<*(lower_bound A)*> ^ D1) . m by A4, A19, XXREAL_0:2; ::_thesis: verum
end;
suppose ex i being Nat st
( i in dom D1 & n = (len <*(lower_bound A)*>) + i ) ; ::_thesis: (<*(lower_bound A)*> ^ D1) . n < (<*(lower_bound A)*> ^ D1) . m
then consider i being Element of NAT such that
A23: i in dom D1 and
A24: n = (len <*(lower_bound A)*>) + i ;
A25: D1 . i = (<*(lower_bound A)*> ^ D1) . n by A23, A24, FINSEQ_1:def_7;
consider j being 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 = (<*(lower_bound A)*> ^ D1) . m by A26, A27, FINSEQ_1:def_7;
i < j by A7, A24, A27, XREAL_1:6;
hence (<*(lower_bound A)*> ^ D1) . n < (<*(lower_bound A)*> ^ D1) . m by A23, A26, A25, A28, SEQM_3:def_1; ::_thesis: verum
end;
end;
end;
hence (<*(lower_bound A)*> ^ D1) . n < (<*(lower_bound A)*> ^ D1) . m ; ::_thesis: verum
end;
hence <*(lower_bound A)*> ^ D1 is non empty increasing FinSequence of REAL by SEQM_3:def_1; ::_thesis: verum
end;
Lm9: for A being non empty closed_interval Subset of REAL
for D2 being Division of A st lower_bound A < D2 . 1 holds
<*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D2 being Division of A st lower_bound A < D2 . 1 holds
<*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL
let D2 be Division of A; ::_thesis: ( lower_bound A < D2 . 1 implies <*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL )
set MD2 = <*(lower_bound A)*> ^ D2;
assume A1: lower_bound A < D2 . 1 ; ::_thesis: <*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL
for n, m being Element of NAT st n in dom (<*(lower_bound A)*> ^ D2) & m in dom (<*(lower_bound A)*> ^ D2) & n < m holds
(<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m
proof
let n, m be Element of NAT ; ::_thesis: ( n in dom (<*(lower_bound A)*> ^ D2) & m in dom (<*(lower_bound A)*> ^ D2) & n < m implies (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m )
assume that
A2: n in dom (<*(lower_bound A)*> ^ D2) and
A3: m in dom (<*(lower_bound A)*> ^ D2) and
A4: n < m ; ::_thesis: (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m
A5: not m in dom <*(lower_bound A)*>
proof
assume m in dom <*(lower_bound A)*> ; ::_thesis: contradiction
then m in Seg (len <*(lower_bound A)*>) by FINSEQ_1:def_3;
then m in {1} by FINSEQ_1:2, FINSEQ_1:39;
then A6: n < 1 by A4, TARSKI:def_1;
n in Seg (len (<*(lower_bound A)*> ^ D2)) by A2, FINSEQ_1:def_3;
hence contradiction by A6, FINSEQ_1:1; ::_thesis: verum
end;
A7: not (<*(lower_bound A)*> ^ D2) . m in rng <*(lower_bound A)*>
proof
assume (<*(lower_bound A)*> ^ D2) . m in rng <*(lower_bound A)*> ; ::_thesis: contradiction
then (<*(lower_bound A)*> ^ D2) . m in {(lower_bound A)} by FINSEQ_1:38;
then A8: (<*(lower_bound A)*> ^ D2) . m = lower_bound A by TARSKI:def_1;
rng D2 <> {} ;
then A9: 1 in dom D2 by FINSEQ_3:32;
consider n being 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 = (<*(lower_bound A)*> ^ D2) . m by A10, A11, FINSEQ_1:def_7;
hence contradiction by A1, A8, A10, A12, A9, SEQ_4:137; ::_thesis: verum
end;
(<*(lower_bound A)*> ^ D2) . m in rng (<*(lower_bound A)*> ^ D2) by A3, FUNCT_1:def_3;
then (<*(lower_bound A)*> ^ D2) . m in (rng <*(lower_bound A)*>) \/ (rng D2) by FINSEQ_1:31;
then A13: ( (<*(lower_bound A)*> ^ D2) . m in rng <*(lower_bound A)*> or (<*(lower_bound A)*> ^ D2) . m in rng D2 ) by XBOOLE_0:def_3;
now__::_thesis:_(<*(lower_bound_A)*>_^_D2)_._n_<_(<*(lower_bound_A)*>_^_D2)_._m
percases ( n in dom <*(lower_bound A)*> or ex i being Nat st
( i in dom D2 & n = (len <*(lower_bound A)*>) + i ) ) by A2, FINSEQ_1:25;
supposeA14: n in dom <*(lower_bound A)*> ; ::_thesis: (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m
then n in Seg (len <*(lower_bound A)*>) by FINSEQ_1:def_3;
then n in {1} by FINSEQ_1:2, FINSEQ_1:39;
then A15: n = 1 by TARSKI:def_1;
A16: (<*(lower_bound A)*> ^ D2) . n = <*(lower_bound A)*> . n by A14, FINSEQ_1:def_7
.= lower_bound A by A15, FINSEQ_1:def_8 ;
rng D2 <> {} ;
then A17: 1 in dom D2 by FINSEQ_3:32;
consider k being Element of NAT such that
A18: k in dom D2 and
A19: (<*(lower_bound A)*> ^ D2) . 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 <= (<*(lower_bound A)*> ^ D2) . m by A18, A19, A17, SEQ_4:137;
hence (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m by A1, A16, XXREAL_0:2; ::_thesis: verum
end;
suppose ex i being Nat st
( i in dom D2 & n = (len <*(lower_bound A)*>) + i ) ; ::_thesis: (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m
then consider i being Element of NAT such that
A20: i in dom D2 and
A21: n = (len <*(lower_bound A)*>) + i ;
A22: D2 . i = (<*(lower_bound A)*> ^ D2) . n by A20, A21, FINSEQ_1:def_7;
consider j being 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 = (<*(lower_bound A)*> ^ D2) . m by A23, A24, FINSEQ_1:def_7;
i < j by A4, A21, A24, XREAL_1:6;
hence (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m by A20, A23, A22, A25, SEQM_3:def_1; ::_thesis: verum
end;
end;
end;
hence (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m ; ::_thesis: verum
end;
hence <*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL by SEQM_3:def_1; ::_thesis: verum
end;
Lm10: for A being non empty closed_interval Subset of REAL
for D1 being Division of A
for f being Function of A,REAL
for MD1 being Division of A st MD1 = <*(lower_bound A)*> ^ D1 holds
( ( for i being Element of NAT 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 A be non empty closed_interval Subset of REAL; ::_thesis: for D1 being Division of A
for f being Function of A,REAL
for MD1 being Division of A st MD1 = <*(lower_bound A)*> ^ D1 holds
( ( for i being Element of NAT 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 )
let D1 be Division of A; ::_thesis: for f being Function of A,REAL
for MD1 being Division of A st MD1 = <*(lower_bound A)*> ^ D1 holds
( ( for i being Element of NAT 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 )
let f be Function of A,REAL; ::_thesis: for MD1 being Division of A st MD1 = <*(lower_bound A)*> ^ D1 holds
( ( for i being Element of NAT 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 )
let MD1 be Division of A; ::_thesis: ( MD1 = <*(lower_bound A)*> ^ D1 implies ( ( for i being Element of NAT 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 ) )
assume A1: MD1 = <*(lower_bound A)*> ^ D1 ; ::_thesis: ( ( for i being Element of NAT 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 )
thus A2: for i being Element of NAT st i in Seg (len D1) holds
divset (MD1,(i + 1)) = divset (D1,i) ::_thesis: ( upper_volume (f,D1) = (upper_volume (f,MD1)) /^ 1 & lower_volume (f,D1) = (lower_volume (f,MD1)) /^ 1 )
proof
let i be Element of NAT ; ::_thesis: ( i in Seg (len D1) implies divset (MD1,(i + 1)) = divset (D1,i) )
assume A3: i in Seg (len D1) ; ::_thesis: divset (MD1,(i + 1)) = divset (D1,i)
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
percases ( i = 1 or i <> 1 ) ;
supposeA9: i = 1 ; ::_thesis: ( lower_bound (divset (D1,i)) = lower_bound (divset (MD1,(i + 1))) & upper_bound (divset (D1,i)) = upper_bound (divset (MD1,(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 ( lower_bound (divset (D1,i)) = lower_bound (divset (MD1,(i + 1))) & upper_bound (divset (D1,i)) = upper_bound (divset (MD1,(i + 1))) ) by A4, A9, A11, A12, INTEGRA1:def_4; ::_thesis: verum
end;
supposeA13: i <> 1 ; ::_thesis: ( lower_bound (divset (D1,i)) = lower_bound (divset (MD1,(i + 1))) & upper_bound (divset (D1,i)) = upper_bound (divset (MD1,(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 ( lower_bound (divset (D1,i)) = lower_bound (divset (MD1,(i + 1))) & upper_bound (divset (D1,i)) = upper_bound (divset (MD1,(i + 1))) ) by A15; ::_thesis: verum
end;
end;
end;
divset (D1,i) = [.(lower_bound (divset (D1,i))),(upper_bound (divset (D1,i))).] by INTEGRA1:4;
hence divset (MD1,(i + 1)) = divset (D1,i) by A8, INTEGRA1:4; ::_thesis: verum
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 ::_thesis: lower_volume (f,D1) = (lower_volume (f,MD1)) /^ 1
proof
set D2 = D1;
set MD2 = MD1;
rng (upper_volume (f,MD1)) <> {} ;
then 1 in dom (upper_volume (f,MD1)) by FINSEQ_3:32;
then 1 <= len (upper_volume (f,MD1)) by FINSEQ_3:25;
then len ((upper_volume (f,MD1)) /^ 1) = (len (upper_volume (f,MD1))) - 1 by RFINSEQ:def_1
.= (len MD1) - 1 by INTEGRA1:def_6
.= len D1 by A16 ;
then A17: len (upper_volume (f,D1)) = len ((upper_volume (f,MD1)) /^ 1) by INTEGRA1:def_6;
for k being Nat st 1 <= k & k <= len (upper_volume (f,D1)) holds
(upper_volume (f,D1)) . k = ((upper_volume (f,MD1)) /^ 1) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len (upper_volume (f,D1)) implies (upper_volume (f,D1)) . k = ((upper_volume (f,MD1)) /^ 1) . k )
assume that
A18: 1 <= k and
A19: k <= len (upper_volume (f,D1)) ; ::_thesis: (upper_volume (f,D1)) . k = ((upper_volume (f,MD1)) /^ 1) . k
k + 1 <= (len (upper_volume (f,D1))) + 1 by A19, XREAL_1:6;
then A20: k + 1 <= (len D1) + 1 by INTEGRA1:def_6;
k in Seg (len (upper_volume (f,D1))) by A18, A19, FINSEQ_1:1;
then A21: k in Seg (len D1) by INTEGRA1:def_6;
then k in dom D1 by FINSEQ_1:def_3;
then A22: (upper_volume (f,D1)) . k = (upper_bound (rng (f | (divset (D1,k))))) * (vol (divset (D1,k))) by INTEGRA1:def_6
.= (upper_bound (rng (f | (divset (MD1,(k + 1)))))) * (vol (divset (D1,k))) by A2, A21
.= (upper_bound (rng (f | (divset (MD1,(k + 1)))))) * (vol (divset (MD1,(k + 1)))) by A2, A21 ;
A23: len ((upper_volume (f,MD1)) /^ 1) <= len (upper_volume (f,MD1)) by FINSEQ_5:25;
1 <= k + 1 by NAT_1:11;
then k + 1 in Seg (len MD1) by A16, A20, FINSEQ_1:1;
then A24: k + 1 in dom MD1 by FINSEQ_1:def_3;
1 <= len (upper_volume (f,D1)) by A18, A19, XXREAL_0:2;
then A25: 1 <= len (upper_volume (f,MD1)) by A17, A23, XXREAL_0:2;
k in dom ((upper_volume (f,MD1)) /^ 1) by A17, A18, A19, FINSEQ_3:25;
then ((upper_volume (f,MD1)) /^ 1) . k = (upper_volume (f,MD1)) . (k + 1) by A25, RFINSEQ:def_1
.= (upper_bound (rng (f | (divset (MD1,(k + 1)))))) * (vol (divset (MD1,(k + 1)))) by A24, INTEGRA1:def_6 ;
hence (upper_volume (f,D1)) . k = ((upper_volume (f,MD1)) /^ 1) . k by A22; ::_thesis: verum
end;
hence upper_volume (f,D1) = (upper_volume (f,MD1)) /^ 1 by A17, FINSEQ_1:14; ::_thesis: verum
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 being Nat st 1 <= k & k <= len (lower_volume (f,D1)) holds
(lower_volume (f,D1)) . k = ((lower_volume (f,MD1)) /^ 1) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len (lower_volume (f,D1)) implies (lower_volume (f,D1)) . k = ((lower_volume (f,MD1)) /^ 1) . k )
assume that
A27: 1 <= k and
A28: k <= len (lower_volume (f,D1)) ; ::_thesis: (lower_volume (f,D1)) . k = ((lower_volume (f,MD1)) /^ 1) . k
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 (lower_volume (f,D1)) . k = ((lower_volume (f,MD1)) /^ 1) . k by A31; ::_thesis: verum
end;
hence lower_volume (f,D1) = (lower_volume (f,MD1)) /^ 1 by A26, FINSEQ_1:14; ::_thesis: verum
end;
Lm11: for A being non empty closed_interval Subset of REAL
for D2, MD2 being Division of A st MD2 = <*(lower_bound A)*> ^ D2 holds
vol (divset (MD2,1)) = 0
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D2, MD2 being Division of A st MD2 = <*(lower_bound A)*> ^ D2 holds
vol (divset (MD2,1)) = 0
let D2, MD2 be Division of A; ::_thesis: ( MD2 = <*(lower_bound A)*> ^ D2 implies vol (divset (MD2,1)) = 0 )
assume A1: MD2 = <*(lower_bound A)*> ^ D2 ; ::_thesis: vol (divset (MD2,1)) = 0
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 vol (divset (MD2,1)) = 0 ; ::_thesis: verum
end;
Lm12: for A being non empty closed_interval Subset of REAL
for D1, MD1 being Division of A st MD1 = <*(lower_bound A)*> ^ D1 holds
delta MD1 = delta D1
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, MD1 being Division of A st MD1 = <*(lower_bound A)*> ^ D1 holds
delta MD1 = delta D1
let D1, MD1 be Division of A; ::_thesis: ( MD1 = <*(lower_bound A)*> ^ D1 implies delta MD1 = delta D1 )
assume A1: MD1 = <*(lower_bound A)*> ^ D1 ; ::_thesis: delta MD1 = delta 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 being Element of NAT 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 being Element of NAT 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
percases ( j = 1 or j <> 1 ) ;
suppose j = 1 ; ::_thesis: delta MD1 <= delta D1
hence delta MD1 <= delta D1 by A2, A9, Th9; ::_thesis: verum
end;
suppose j <> 1 ; ::_thesis: delta MD1 <= delta D1
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 being 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 delta MD1 <= delta D1 by XXREAL_2:def_8; ::_thesis: verum
end;
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 delta MD1 = delta D1 by A10, XXREAL_0:1; ::_thesis: verum
end;
theorem Th13: :: INTEGRA3:13
for x being Real
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st x in divset (D1,(len D1)) & vol A <> 0 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & f | A is bounded & x > lower_bound A holds
(Sum (lower_volume (f,D2))) - (Sum (lower_volume (f,D1))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
proof
let x be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st x in divset (D1,(len D1)) & vol A <> 0 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & f | A is bounded & x > lower_bound A holds
(Sum (lower_volume (f,D2))) - (Sum (lower_volume (f,D1))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for f being Function of A,REAL st x in divset (D1,(len D1)) & vol A <> 0 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & f | A is bounded & x > lower_bound A holds
(Sum (lower_volume (f,D2))) - (Sum (lower_volume (f,D1))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
let D1, D2 be Division of A; ::_thesis: for f being Function of A,REAL st x in divset (D1,(len D1)) & vol A <> 0 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & f | A is bounded & x > lower_bound A holds
(Sum (lower_volume (f,D2))) - (Sum (lower_volume (f,D1))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
let f be Function of A,REAL; ::_thesis: ( 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) )
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 ; ::_thesis: (Sum (lower_volume (f,D2))) - (Sum (lower_volume (f,D1))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
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__::_thesis:_(Sum_(lower_volume_(f,D2)))_-_(Sum_(lower_volume_(f,D1)))_<=_((upper_bound_(rng_f))_-_(lower_bound_(rng_f)))_*_(delta_D1)
percases ( len D1 = 1 or len D1 >= 2 ) by A8;
supposeA9: len D1 = 1 ; ::_thesis: (Sum (lower_volume (f,D2))) - (Sum (lower_volume (f,D1))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
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 Real st y in rng MD1 holds
y in A
proof
let y be Real; ::_thesis: ( y in rng MD1 implies y in A )
assume y in rng MD1 ; ::_thesis: y in A
then A12: y in (rng <*(lower_bound A)*>) \/ (rng D1) by FINSEQ_1:31;
percases ( y in rng <*(lower_bound A)*> or y in rng D1 ) by A12, XBOOLE_0:def_3;
suppose y in rng <*(lower_bound A)*> ; ::_thesis: y in A
then y in {(lower_bound A)} by FINSEQ_1:38;
then A13: y = lower_bound A by TARSKI:def_1;
ex a, b being Real st
( a <= b & a = lower_bound A & b = upper_bound A ) by SEQ_4:11;
hence y in A by A13, INTEGRA2:1; ::_thesis: verum
end;
supposeA14: y in rng D1 ; ::_thesis: y in A
rng D1 c= A by INTEGRA1:def_2;
hence y in A by A14; ::_thesis: verum
end;
end;
end;
then rng MD1 c= A by SUBSET_1:2;
then reconsider MD1 = 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
percases ( D2 . 1 in rng D1 or D2 . 1 in {x} ) by A4, A29, XBOOLE_0:def_3;
supposeA30: D2 . 1 in rng D1 ; ::_thesis: lower_bound A < D2 . 1
rng D1 <> {} ;
then A31: 1 in dom D1 by FINSEQ_3:32;
consider k being Element of NAT 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 lower_bound A < D2 . 1 by A17, XXREAL_0:2; ::_thesis: verum
end;
suppose D2 . 1 in {x} ; ::_thesis: lower_bound A < D2 . 1
hence lower_bound A < D2 . 1 by A6, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
then reconsider MD2 = <*(lower_bound A)*> ^ D2 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 Real st y in rng MD2 holds
y in A
proof
let y be Real; ::_thesis: ( y in rng MD2 implies y in A )
assume y in rng MD2 ; ::_thesis: y in A
then A35: y in (rng <*(lower_bound A)*>) \/ (rng D2) by FINSEQ_1:31;
percases ( y in rng <*(lower_bound A)*> or y in rng D2 ) by A35, XBOOLE_0:def_3;
suppose y in rng <*(lower_bound A)*> ; ::_thesis: y in A
then y in {(lower_bound A)} by FINSEQ_1:38;
then A36: y = lower_bound A by TARSKI:def_1;
ex a, b being Real st
( a <= b & a = lower_bound A & b = upper_bound A ) by SEQ_4:11;
hence y in A by A36, INTEGRA2:1; ::_thesis: verum
end;
supposeA37: y in rng D2 ; ::_thesis: y in A
rng D2 c= A by INTEGRA1:def_2;
hence y in A by A37; ::_thesis: verum
end;
end;
end;
then rng MD2 c= A by SUBSET_1:2;
then reconsider MD2 = 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 (Sum (lower_volume (f,D2))) - (Sum (lower_volume (f,D1))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) by A43, A18, A48, A47, Lm12; ::_thesis: verum
end;
suppose len D1 >= 2 ; ::_thesis: (Sum (lower_volume (f,D2))) - (Sum (lower_volume (f,D1))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
hence (Sum (lower_volume (f,D2))) - (Sum (lower_volume (f,D1))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) by A1, A3, A4, A5, Th10; ::_thesis: verum
end;
end;
end;
hence (Sum (lower_volume (f,D2))) - (Sum (lower_volume (f,D1))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) ; ::_thesis: verum
end;
theorem Th14: :: INTEGRA3:14
for x being Real
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st x in divset (D1,(len D1)) & vol A <> 0 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & f | A is bounded & x > lower_bound A holds
(Sum (upper_volume (f,D1))) - (Sum (upper_volume (f,D2))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
proof
let x be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st x in divset (D1,(len D1)) & vol A <> 0 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & f | A is bounded & x > lower_bound A holds
(Sum (upper_volume (f,D1))) - (Sum (upper_volume (f,D2))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for f being Function of A,REAL st x in divset (D1,(len D1)) & vol A <> 0 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & f | A is bounded & x > lower_bound A holds
(Sum (upper_volume (f,D1))) - (Sum (upper_volume (f,D2))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
let D1, D2 be Division of A; ::_thesis: for f being Function of A,REAL st x in divset (D1,(len D1)) & vol A <> 0 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & f | A is bounded & x > lower_bound A holds
(Sum (upper_volume (f,D1))) - (Sum (upper_volume (f,D2))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
let f be Function of A,REAL; ::_thesis: ( 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) )
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 ; ::_thesis: (Sum (upper_volume (f,D1))) - (Sum (upper_volume (f,D2))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
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__::_thesis:_(Sum_(upper_volume_(f,D1)))_-_(Sum_(upper_volume_(f,D2)))_<=_((upper_bound_(rng_f))_-_(lower_bound_(rng_f)))_*_(delta_D1)
percases ( len D1 = 1 or len D1 >= 2 ) by A8;
supposeA9: len D1 = 1 ; ::_thesis: (Sum (upper_volume (f,D1))) - (Sum (upper_volume (f,D2))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
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 Real st y in rng MD1 holds
y in A
proof
let y be Real; ::_thesis: ( y in rng MD1 implies y in A )
assume y in rng MD1 ; ::_thesis: y in A
then A12: y in (rng <*(lower_bound A)*>) \/ (rng D1) by FINSEQ_1:31;
percases ( y in rng <*(lower_bound A)*> or y in rng D1 ) by A12, XBOOLE_0:def_3;
suppose y in rng <*(lower_bound A)*> ; ::_thesis: y in A
then y in {(lower_bound A)} by FINSEQ_1:38;
then A13: y = lower_bound A by TARSKI:def_1;
ex a, b being Real st
( a <= b & a = lower_bound A & b = upper_bound A ) by SEQ_4:11;
hence y in A by A13, INTEGRA2:1; ::_thesis: verum
end;
supposeA14: y in rng D1 ; ::_thesis: y in A
rng D1 c= A by INTEGRA1:def_2;
hence y in A by A14; ::_thesis: verum
end;
end;
end;
then rng MD1 c= A by SUBSET_1:2;
then reconsider MD1 = 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
percases ( D2 . 1 in rng D1 or D2 . 1 in {x} ) by A4, A29, XBOOLE_0:def_3;
supposeA30: D2 . 1 in rng D1 ; ::_thesis: lower_bound A < D2 . 1
rng D1 <> {} ;
then A31: 1 in dom D1 by FINSEQ_3:32;
consider k being Element of NAT 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 lower_bound A < D2 . 1 by A17, XXREAL_0:2; ::_thesis: verum
end;
suppose D2 . 1 in {x} ; ::_thesis: lower_bound A < D2 . 1
hence lower_bound A < D2 . 1 by A6, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
then reconsider MD2 = <*(lower_bound A)*> ^ D2 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 Real st y in rng MD2 holds
y in A
proof
let y be Real; ::_thesis: ( y in rng MD2 implies y in A )
assume y in rng MD2 ; ::_thesis: y in A
then A35: y in (rng <*(lower_bound A)*>) \/ (rng D2) by FINSEQ_1:31;
percases ( y in rng <*(lower_bound A)*> or y in rng D2 ) by A35, XBOOLE_0:def_3;
suppose y in rng <*(lower_bound A)*> ; ::_thesis: y in A
then y in {(lower_bound A)} by FINSEQ_1:38;
then A36: y = lower_bound A by TARSKI:def_1;
ex a, b being Real st
( a <= b & a = lower_bound A & b = upper_bound A ) by SEQ_4:11;
hence y in A by A36, INTEGRA2:1; ::_thesis: verum
end;
supposeA37: y in rng D2 ; ::_thesis: y in A
rng D2 c= A by INTEGRA1:def_2;
hence y in A by A37; ::_thesis: verum
end;
end;
end;
then rng MD2 c= A by SUBSET_1:2;
then reconsider MD2 = 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 (Sum (upper_volume (f,D1))) - (Sum (upper_volume (f,D2))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) by A43, A18, A48, A47, Lm12; ::_thesis: verum
end;
suppose len D1 >= 2 ; ::_thesis: (Sum (upper_volume (f,D1))) - (Sum (upper_volume (f,D2))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
hence (Sum (upper_volume (f,D1))) - (Sum (upper_volume (f,D2))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) by A1, A3, A4, A5, Th11; ::_thesis: verum
end;
end;
end;
hence (Sum (upper_volume (f,D1))) - (Sum (upper_volume (f,D2))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) ; ::_thesis: verum
end;
theorem Th15: :: INTEGRA3:15
for r being Real
for i, j being Element of NAT
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A st i in dom D1 & j in dom D1 & i <= j & D1 <= D2 & r < (mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j)))) . 1 holds
ex B being non empty closed_interval Subset of REAL ex MD1, MD2 being 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
let r be Real; ::_thesis: for i, j being Element of NAT
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A st i in dom D1 & j in dom D1 & i <= j & D1 <= D2 & r < (mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j)))) . 1 holds
ex B being non empty closed_interval Subset of REAL ex MD1, MD2 being 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))) )
let i, j be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A st i in dom D1 & j in dom D1 & i <= j & D1 <= D2 & r < (mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j)))) . 1 holds
ex B being non empty closed_interval Subset of REAL ex MD1, MD2 being 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))) )
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A st i in dom D1 & j in dom D1 & i <= j & D1 <= D2 & r < (mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j)))) . 1 holds
ex B being non empty closed_interval Subset of REAL ex MD1, MD2 being 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))) )
let D1, D2 be Division of A; ::_thesis: ( 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 being non empty closed_interval Subset of REAL ex MD1, MD2 being 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))) ) )
set MD1 = mid (D1,i,j);
set MD2 = mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j)));
assume A1: i in dom D1 ; ::_thesis: ( not j in dom D1 or not i <= j or not D1 <= D2 or not r < (mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j)))) . 1 or ex B being non empty closed_interval Subset of REAL ex MD1, MD2 being 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))) ) )
then A2: 1 <= i by FINSEQ_3:25;
assume A3: j in dom D1 ; ::_thesis: ( not i <= j or not D1 <= D2 or not r < (mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j)))) . 1 or ex B being non empty closed_interval Subset of REAL ex MD1, MD2 being 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))) ) )
assume A4: i <= j ; ::_thesis: ( not D1 <= D2 or not r < (mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j)))) . 1 or ex B being non empty closed_interval Subset of REAL ex MD1, MD2 being 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))) ) )
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: (mid (D1,i,j)) . 1 = D1 . ((1 + i) - 1) by A4, A5, A2, FINSEQ_6:122
.= D1 . i ;
assume A8: D1 <= D2 ; ::_thesis: ( not r < (mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j)))) . 1 or ex B being non empty closed_interval Subset of REAL ex MD1, MD2 being 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))) ) )
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 ; ::_thesis: ex B being non empty closed_interval Subset of REAL ex MD1, MD2 being 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))) )
then consider B being non empty closed_interval Subset of REAL such that
A17: r = lower_bound B and
A18: upper_bound B = (mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j)))) . (len (mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j))))) and
A19: mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j))) is Division of B by A11, A13, A15, Th12;
A20: len (mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j)))) = ((indx (D2,D1,j)) - (indx (D2,D1,i))) + 1 by A11, A13, A15, INTEGRA1:58;
reconsider MD2 = mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j))) 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 = (mid (D1,i,j)) . (len (mid (D1,i,j))) and
A25: mid (D1,i,j) is Division of C by A1, A3, A4, A16, A7, Th12;
len (mid (D1,i,j)) = (j - i) + 1 by A1, A3, A4, INTEGRA1:58;
then A26: (mid (D1,i,j)) . (len (mid (D1,i,j))) = 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 = mid (D1,i,j) as Division of B by A25;
A28: rng MD1 c= rng MD2
proof
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in rng MD1 or x1 in rng MD2 )
A29: rng MD1 c= rng D1 by FINSEQ_6:119;
assume A30: x1 in rng MD1 ; ::_thesis: x1 in rng MD2
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, XBOOLE_1:1;
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, NAT_1:12;
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;
k3 in NAT by ORDINAL1:def_12;
then MD2 . k3 = D2 . ((k3 + (indx (D2,D1,i))) - 1) by A15, A12, A14, A40, A39, A38, FINSEQ_6:122;
hence x1 in rng MD2 by A34, A38, A41, FUNCT_1:def_3; ::_thesis: verum
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 ex B being non empty closed_interval Subset of REAL ex MD1, MD2 being 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))) ) by A17, A18, A24, A27; ::_thesis: verum
end;
theorem Th16: :: INTEGRA3:16
for x being Real
for A being non empty closed_interval Subset of REAL
for D being Division of A st x in rng D holds
( D . 1 <= x & x <= D . (len D) )
proof
let x be Real; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A st x in rng D holds
( D . 1 <= x & x <= D . (len D) )
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A st x in rng D holds
( D . 1 <= x & x <= D . (len D) )
let D be Division of A; ::_thesis: ( x in rng D implies ( D . 1 <= x & x <= D . (len D) ) )
assume x in rng D ; ::_thesis: ( D . 1 <= x & x <= D . (len D) )
then consider i being Element of NAT 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 ( D . 1 <= x & x <= D . (len D) ) by A1, A2, A4, A3, A6, SEQ_4:137; ::_thesis: verum
end;
theorem Th17: :: INTEGRA3:17
for p being FinSequence of REAL
for i, j, k being Element of NAT 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 ; ::_thesis: for i, j, k being Element of NAT 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))
let i, j, k be Element of NAT ; ::_thesis: ( p is increasing & i in dom p & j in dom p & k in dom p & p . i <= p . k & p . k <= p . j implies p . k in rng (mid (p,i,j)) )
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 ; ::_thesis: p . k in rng (mid (p,i,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, NAT_1:12;
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;
n in NAT by ORDINAL1:def_12;
then (mid (p,i,j)) . n = p . ((n + i) - 1) by A7, A9, A15, A11, A14, A13, FINSEQ_6:122
.= p . k by A11 ;
hence p . k in rng (mid (p,i,j)) by A16, FUNCT_1:def_3; ::_thesis: verum
end;
theorem Th18: :: INTEGRA3:18
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded & i in dom D holds
lower_bound (rng (f | (divset (D,i)))) <= upper_bound (rng f)
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded & i in dom D holds
lower_bound (rng (f | (divset (D,i)))) <= upper_bound (rng f)
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A
for f being Function of A,REAL st f | A is bounded & i in dom D holds
lower_bound (rng (f | (divset (D,i)))) <= upper_bound (rng f)
let D be Division of A; ::_thesis: for f being Function of A,REAL st f | A is bounded & i in dom D holds
lower_bound (rng (f | (divset (D,i)))) <= upper_bound (rng f)
let f be Function of A,REAL; ::_thesis: ( f | A is bounded & i in dom D implies lower_bound (rng (f | (divset (D,i)))) <= upper_bound (rng f) )
assume A1: f | A is bounded ; ::_thesis: ( not i in dom D or lower_bound (rng (f | (divset (D,i)))) <= upper_bound (rng f) )
assume i in dom D ; ::_thesis: lower_bound (rng (f | (divset (D,i)))) <= upper_bound (rng f)
then divset (D,i) c= A by INTEGRA1:8;
hence lower_bound (rng (f | (divset (D,i)))) <= upper_bound (rng f) by A1, Lm4; ::_thesis: verum
end;
theorem Th19: :: INTEGRA3:19
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded & i in dom D holds
upper_bound (rng (f | (divset (D,i)))) >= lower_bound (rng f)
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded & i in dom D holds
upper_bound (rng (f | (divset (D,i)))) >= lower_bound (rng f)
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A
for f being Function of A,REAL st f | A is bounded & i in dom D holds
upper_bound (rng (f | (divset (D,i)))) >= lower_bound (rng f)
let D be Division of A; ::_thesis: for f being Function of A,REAL st f | A is bounded & i in dom D holds
upper_bound (rng (f | (divset (D,i)))) >= lower_bound (rng f)
let f be Function of A,REAL; ::_thesis: ( f | A is bounded & i in dom D implies upper_bound (rng (f | (divset (D,i)))) >= lower_bound (rng f) )
assume A1: f | A is bounded ; ::_thesis: ( not i in dom D or upper_bound (rng (f | (divset (D,i)))) >= lower_bound (rng f) )
assume i in dom D ; ::_thesis: upper_bound (rng (f | (divset (D,i)))) >= lower_bound (rng f)
then divset (D,i) c= A by INTEGRA1:8;
hence upper_bound (rng (f | (divset (D,i)))) >= lower_bound (rng f) by A1, Lm4; ::_thesis: verum
end;
begin
theorem :: INTEGRA3:20
for A being non empty closed_interval Subset of REAL
for f being Function of A,REAL
for T being DivSequence of A st f | A is bounded & delta T is 0 -convergent & delta T is non-zero & vol A <> 0 holds
( lower_sum (f,T) is convergent & lim (lower_sum (f,T)) = lower_integral f )
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being Function of A,REAL
for T being DivSequence of A st f | A is bounded & delta T is 0 -convergent & delta T is non-zero & vol A <> 0 holds
( lower_sum (f,T) is convergent & lim (lower_sum (f,T)) = lower_integral f )
let f be Function of A,REAL; ::_thesis: for T being DivSequence of A st f | A is bounded & delta T is 0 -convergent & delta T is non-zero & vol A <> 0 holds
( lower_sum (f,T) is convergent & lim (lower_sum (f,T)) = lower_integral f )
let T be DivSequence of A; ::_thesis: ( f | A is bounded & delta T is 0 -convergent & delta T is non-zero & vol A <> 0 implies ( lower_sum (f,T) is convergent & lim (lower_sum (f,T)) = lower_integral f ) )
assume that
A1: f | A is bounded and
A2: ( delta T is 0 -convergent & delta T is non-zero ) and
A3: vol A <> 0 ; ::_thesis: ( lower_sum (f,T) is convergent & lim (lower_sum (f,T)) = lower_integral f )
A4: delta T is convergent by A2, FDIFF_1:def_1;
A5: for D, D1 being Division of A ex D2 being Division of A 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 be Division of A; ::_thesis: ex D2 being Division of A 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)) )
consider D2 being Division of A 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 ex D2 being Division of A 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 A6, A7, A8, A9; ::_thesis: verum
end;
A10: for D, D1 being Division of A st delta D1 < min (rng (upper_volume ((chi (A,A)),D))) holds
ex D2 being Division of A 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 be Division of A; ::_thesis: ( delta D1 < min (rng (upper_volume ((chi (A,A)),D))) implies ex D2 being Division of A 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) ) )
assume A11: delta D1 < min (rng (upper_volume ((chi (A,A)),D))) ; ::_thesis: ex D2 being Division of A 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) )
ex D2 being Division of A 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 being Division of A 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 H1( Division of A) -> FinSequence of REAL = lower_volume (f,$1);
deffunc H2( Division of A, Nat) -> Element of REAL = (PartSums (lower_volume (f,$1))) . $2;
A15: len D2 in dom D2 by FINSEQ_5:6;
A16: for i being Element of NAT st i in dom D holds
ex j being Element of NAT st
( j in dom D1 & D . i in divset (D1,j) & H2(D2, indx (D2,D1,j)) - H2(D1,j) <= (i * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) )
proof
defpred S1[ non empty Nat] means ( $1 in dom D implies ex j being Element of NAT st
( j in dom D1 & D . $1 in divset (D1,j) & H2(D2, indx (D2,D1,j)) - H2(D1,j) <= ($1 * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) ) );
let i be Element of NAT ; ::_thesis: ( i in dom D implies ex j being Element of NAT st
( j in dom D1 & D . i in divset (D1,j) & H2(D2, indx (D2,D1,j)) - H2(D1,j) <= (i * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) ) )
assume A17: i in dom D ; ::_thesis: ex j being Element of NAT st
( j in dom D1 & D . i in divset (D1,j) & H2(D2, indx (D2,D1,j)) - H2(D1,j) <= (i * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) )
then A18: i in Seg (len D) by FINSEQ_1:def_3;
A19: for i, j being Element of NAT st i in dom D & j in dom D1 & D . i in divset (D1,j) holds
j >= 2
proof
let i, j be Element of NAT ; ::_thesis: ( i in dom D & j in dom D1 & D . i in divset (D1,j) implies j >= 2 )
assume A20: i in dom D ; ::_thesis: ( not j in dom D1 or not D . i in divset (D1,j) or j >= 2 )
assume that
A21: j in dom D1 and
A22: D . i in divset (D1,j) ; ::_thesis: j >= 2
assume j < 2 ; ::_thesis: contradiction
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
percases ( i = 1 or i <> 1 ) ;
supposeA26: i = 1 ; ::_thesis: delta D1 >= min (rng (upper_volume ((chi (A,A)),D)))
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 delta D1 >= min (rng (upper_volume ((chi (A,A)),D))) by A31, XXREAL_0:2; ::_thesis: verum
end;
supposeA34: i <> 1 ; ::_thesis: delta D1 >= min (rng (upper_volume ((chi (A,A)),D)))
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 delta D1 >= min (rng (upper_volume ((chi (A,A)),D))) by A39, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence contradiction by A11; ::_thesis: verum
end;
A40: S1[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 being Element of NAT such that
A42: j in dom D1 and
A43: D . 1 in divset (D1,j) by Th3, INTEGRA1:6;
H2(D2, indx (D2,D1,j)) - H2(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 ; ::_thesis: contradiction
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 ) ; ::_thesis: contradiction
percases ( D2 . ID1 in rng D1 or D2 . ID2 in rng D1 ) by A70;
suppose D2 . ID1 in rng D1 ; ::_thesis: contradiction
then consider n being Element of NAT 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; ::_thesis: verum
end;
suppose D2 . ID2 in rng D1 ; ::_thesis: contradiction
then consider n being Element of NAT 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; ::_thesis: verum
end;
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; ::_thesis: verum
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;
(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) = <*((lower_volume (f,D1)) . j)*> 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__::_thesis:_(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)
percases ( len (mid ((lower_volume (f,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) = 1 or len (mid ((lower_volume (f,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) = 2 ) by A94, A93, Lm2;
supposeA95: len (mid ((lower_volume (f,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) = 1 ; ::_thesis: (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)
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;
(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))) = <*((lower_volume (f,D2)) . ((indx (D2,D1,j1)) + 1))*> 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 (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) by A98, A99; ::_thesis: verum
end;
supposeA104: len (mid ((lower_volume (f,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) = 2 ; ::_thesis: (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)
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 = H1(D2) . ((2 + ((indx (D2,D1,j1)) + 1)) -' 1) by A53, A88, A89, A82, A90, A104, FINSEQ_6:118
.= H1(D2) . ((2 + ((indx (D2,D1,j1)) + 1)) - 1) by A106, XREAL_1:233
.= H1(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 (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) - (Sum (mid (H1(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,((indx (D2,D1,j1)) + 1)));
set C = vol (divset (D2,((indx (D2,D1,j1)) + 2)));
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;
(indx (D2,D1,j1)) + 1 in dom D2 by A91, FINSEQ_1:def_3;
then divset (D2,((indx (D2,D1,j1)) + 1)) c= A by INTEGRA1:8;
then lower_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 1))))) <= upper_bound (rng f) by A1, Lm4;
then A124: (lower_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) <= (upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) 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,((indx (D2,D1,j1)) + 2)) c= A by A125, INTEGRA1:8;
then A127: lower_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 2))))) <= upper_bound (rng f) by A1, Lm4;
reconsider A = lower_bound (rng (f | (divset (D1,j)))) as real number ;
A128: ((lower_volume (f,D1)) . j) - (A * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) = A * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) by A121;
(lower_bound (rng (f | (divset (D1,j))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) >= (lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) by A110, A123, XREAL_1:64;
then Sum (mid (H1(D1),j,j)) >= ((lower_bound (rng (f | (divset (D1,j))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) + ((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) by A86, A128, XREAL_1:19;
then A129: (Sum (mid (H1(D1),j,j))) - ((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) >= (lower_bound (rng (f | (divset (D1,j))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by XREAL_1:19;
(lower_bound (rng (f | (divset (D1,j))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) >= (lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A108, A123, XREAL_1:64;
then (Sum (mid (H1(D1),j,j))) - ((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) >= (lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A129, XXREAL_0:2;
then A130: Sum (mid (H1(D1),j,j)) >= ((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) by XREAL_1:19;
Sum (mid (H1(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))))) + (H1(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 (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) - ((lower_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) <= (upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) by A110, A127, XREAL_1:64;
then Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) <= ((upper_bound (rng f)) * (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 XREAL_1:20;
then (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) - ((upper_bound (rng f)) * (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 XREAL_1:20;
then (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) - ((upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) <= (upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A124, XXREAL_0:2;
then Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) <= ((upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) by XREAL_1:20;
then (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) - (Sum (mid (H1(D1),j,j))) <= (((upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))))) - (((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))))) by A130, XREAL_1:13;
hence (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) - (Sum (mid (H1(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))))) ; ::_thesis: verum
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 (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) by A120, A122, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (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) ; ::_thesis: verum
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 set st x1 in rng (D1 | j1) holds
x1 in rng (D2 | (indx (D2,D1,j1)))
proof
let x1 be set ; ::_thesis: ( x1 in rng (D1 | j1) implies x1 in rng (D2 | (indx (D2,D1,j1))) )
assume x1 in rng (D1 | j1) ; ::_thesis: x1 in rng (D2 | (indx (D2,D1,j1)))
then consider k being Element of NAT 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 being Element of NAT 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 x1 in rng (D2 | (indx (D2,D1,j1))) by A145, A148, FUNCT_1:def_3; ::_thesis: verum
end;
then A149: rng (D1 | j1) c= rng (D2 | (indx (D2,D1,j1))) by TARSKI:def_3;
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 set st x1 in rng (D2 | (indx (D2,D1,j1))) holds
x1 in rng (D1 | j1)
proof
let x1 be set ; ::_thesis: ( x1 in rng (D2 | (indx (D2,D1,j1))) implies x1 in rng (D1 | j1) )
assume x1 in rng (D2 | (indx (D2,D1,j1))) ; ::_thesis: x1 in rng (D1 | j1)
then consider k being Element of NAT 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 ; ::_thesis: D2 . k in rng (D1 | j1)
then consider m being Element of NAT 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 D2 . k in rng (D1 | j1) by A163, FUNCT_1:def_3; ::_thesis: verum
end;
A164: ( D2 . k in rng D implies D2 . k = D1 . j1 )
proof
assume D2 . k in rng D ; ::_thesis: D2 . k = D1 . j1
then consider n being Element of NAT 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 D2 . k = D1 . j1 by A157, XXREAL_0:1; ::_thesis: verum
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 ; ::_thesis: D2 . k in rng (D1 | j1)
j1 in Seg j1 by A150, FINSEQ_1:1;
hence D2 . k in rng (D1 | j1) by A45, A164, A169, A168, RFINSEQ:6; ::_thesis: verum
end;
D2 . k in rng D2 by A155, FUNCT_1:def_3;
hence x1 in rng (D1 | j1) by A14, A48, A153, A154, A167, A158, RFINSEQ:6, XBOOLE_0:def_3; ::_thesis: verum
end;
then rng (D2 | (indx (D2,D1,j1))) c= rng (D1 | j1) by TARSKI:def_3;
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, INTEGRA1:35;
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 being Element of NAT st 1 <= k & k <= j1 holds
k = indx (D2,D1,k)
proof
let k be Element of NAT ; ::_thesis: ( 1 <= k & k <= j1 implies k = indx (D2,D1,k) )
assume that
A174: 1 <= k and
A175: k <= j1 ; ::_thesis: k = indx (D2,D1,k)
assume A176: k <> indx (D2,D1,k) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( k > indx (D2,D1,k) or k < indx (D2,D1,k) ) by A176, XXREAL_0:1;
supposeA177: k > indx (D2,D1,k) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
supposeA184: k < indx (D2,D1,k) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A191: for k being 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; ::_thesis: ( 1 <= k & k <= len ((lower_volume (f,D1)) | j1) implies ((lower_volume (f,D1)) | j1) . k = ((lower_volume (f,D2)) | (indx (D2,D1,j1))) . k )
assume that
A193: 1 <= k and
A194: k <= len ((lower_volume (f,D1)) | j1) ; ::_thesis: ((lower_volume (f,D1)) | j1) . k = ((lower_volume (f,D2)) | (indx (D2,D1,j1))) . k
reconsider k = 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
percases ( k = 1 or k <> 1 ) ;
supposeA205: k = 1 ; ::_thesis: ( 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)))) )
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 ( 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)))) ) by A199, A203, A207, A206, INTEGRA1:def_4; ::_thesis: verum
end;
supposeA208: k <> 1 ; ::_thesis: ( 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)))) )
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 ( 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)))) ) by A13, A198, A215, A210, A213, A214, A211, INTEGRA1:def_19; ::_thesis: verum
end;
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 ((lower_volume (f,D1)) | j1) . k = ((lower_volume (f,D2)) | (indx (D2,D1,j1))) . k by A218; ::_thesis: verum
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 H1(D2)) by A132, INTEGRA1:def_7;
then A225: indx (D2,D1,j) in dom H1(D2) by FINSEQ_1:def_3;
indx (D2,D1,j) <= len D2 by A132, FINSEQ_1:1;
then A226: indx (D2,D1,j) <= len H1(D2) by INTEGRA1:def_7;
j in Seg (len H1(D1)) by A223, INTEGRA1:def_7;
then A227: j in dom H1(D1) by FINSEQ_1:def_3;
j <= len D1 by A223, FINSEQ_1:1;
then A228: j <= len H1(D1) by INTEGRA1:def_7;
j1 in Seg (len D1) by A45, FINSEQ_1:def_3;
then j1 in Seg (len H1(D1)) by INTEGRA1:def_7;
then j1 in dom H1(D1) by FINSEQ_1:def_3;
then H2(D1,j1) = Sum (H1(D1) | j1) by INTEGRA1:def_20;
then H2(D1,j1) + (Sum (mid (H1(D1),j,j))) = Sum ((H1(D1) | j1) ^ (mid (H1(D1),j,j))) by RVSUM_1:75
.= Sum ((mid (H1(D1),1,j1)) ^ (mid (H1(D1),(j1 + 1),j))) by A150, FINSEQ_6:116
.= Sum (mid (H1(D1),1,j)) by A150, A228, A131, INTEGRA2:4
.= Sum (H1(D1) | j) by A224, FINSEQ_6:116 ;
then A229: H2(D1,j1) + (Sum (mid ((lower_volume (f,D1)),j,j))) = H2(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 H1(D2)) by INTEGRA1:def_7;
then indx (D2,D1,j1) in dom H1(D2) by FINSEQ_1:def_3;
then H2(D2, indx (D2,D1,j1)) = Sum (H1(D2) | (indx (D2,D1,j1))) by INTEGRA1:def_20;
then H2(D2, indx (D2,D1,j1)) + (Sum (mid ((lower_volume (f,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) = Sum ((H1(D2) | (indx (D2,D1,j1))) ^ (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) by RVSUM_1:75
.= Sum ((mid (H1(D2),1,(indx (D2,D1,j1)))) ^ (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) by A50, FINSEQ_6:116
.= Sum (mid (H1(D2),1,(indx (D2,D1,j)))) by A50, A52, A226, INTEGRA2:4
.= Sum (H1(D2) | (indx (D2,D1,j))) by A133, FINSEQ_6:116 ;
then A230: H2(D2, indx (D2,D1,j1)) + (Sum (mid ((lower_volume (f,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) = H2(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 H2(D2, indx (D2,D1,j1)) = Sum ((lower_volume (f,D2)) | (indx (D2,D1,j1))) by INTEGRA1:def_20
.= H2(D1,j1) by A222, A46, INTEGRA1:def_20 ;
hence H2(D2, indx (D2,D1,j)) - H2(D1,j) <= (1 * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by A54, A230, A229; ::_thesis: verum
end;
hence S1[1] by A42, A43; ::_thesis: verum
end;
reconsider i = i as non empty Element of NAT by A18, FINSEQ_1:1;
A231: for i being non empty Nat st S1[i] holds
S1[i + 1]
proof
let i be non empty Nat; ::_thesis: ( S1[i] implies S1[i + 1] )
A232: i >= 1 by NAT_1:14;
assume A233: S1[i] ; ::_thesis: S1[i + 1]
S1[i + 1]
proof
A234: i <= i + 1 by NAT_1:11;
assume A235: i + 1 in dom D ; ::_thesis: ex j being Element of NAT st
( j in dom D1 & D . (i + 1) in divset (D1,j) & H2(D2, indx (D2,D1,j)) - H2(D1,j) <= ((i + 1) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) )
then consider j being Element of NAT 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: H2(D2, indx (D2,D1,n1)) - H2(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 ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( n1 = j or n1 > j ) by A246, XXREAL_0:1;
supposeA247: n1 = j ; ::_thesis: contradiction
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; ::_thesis: verum
end;
suppose n1 > j ; ::_thesis: contradiction
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
percases ( j = 1 or j <> 1 ) ;
suppose j = 1 ; ::_thesis: upper_bound (divset (D1,j)) = D1 . j
hence upper_bound (divset (D1,j)) = D1 . j by A236, INTEGRA1:def_4; ::_thesis: verum
end;
suppose j <> 1 ; ::_thesis: upper_bound (divset (D1,j)) = D1 . j
hence upper_bound (divset (D1,j)) = D1 . j by A236, INTEGRA1:def_4; ::_thesis: verum
end;
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; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
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 H1(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 H1(D1) by A266, INTEGRA1:def_7;
then j in Seg (len H1(D1)) by A265, FINSEQ_1:1;
then A276: j in dom H1(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 H1(D1)) by INTEGRA1:def_7;
then n1 in dom H1(D1) by FINSEQ_1:def_3;
then H2(D1,n1) = Sum (H1(D1) | n1) by INTEGRA1:def_20
.= Sum (mid (H1(D1),1,n1)) by A258, FINSEQ_6:116 ;
then H2(D1,n1) + (Sum (mid (H1(D1),(n1 + 1),j))) = Sum ((mid (H1(D1),1,n1)) ^ (mid (H1(D1),(n1 + 1),j))) by RVSUM_1:75
.= Sum (mid (H1(D1),1,j)) by A245, A258, A275, INTEGRA2:4
.= Sum (H1(D1) | j) by A265, FINSEQ_6:116 ;
then A278: H2(D1,j) = H2(D1,n1) + (Sum (mid (H1(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 H1(D2)) by INTEGRA1:def_7;
then A279: indx (D2,D1,j) in dom H1(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 (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) - (Sum (mid (H1(D1),(n1 + 1),j))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
proof
now__::_thesis:_(Sum_(mid_(H1(D2),((indx_(D2,D1,n1))_+_1),(indx_(D2,D1,j)))))_-_(Sum_(mid_(H1(D1),(n1_+_1),j)))_<=_((upper_bound_(rng_f))_-_(lower_bound_(rng_f)))_*_(delta_D1)
percases ( n1 + 1 = j or n1 + 1 < j ) by A257, XXREAL_0:1;
supposeA282: n1 + 1 = j ; ::_thesis: (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) - (Sum (mid (H1(D1),(n1 + 1),j))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
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 ; ::_thesis: contradiction
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 ; ::_thesis: contradiction
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; ::_thesis: verum
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 ; ::_thesis: contradiction
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; ::_thesis: verum
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; ::_thesis: verum
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) ; ::_thesis: (indx (D2,D1,n1)) + 2 = 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 (indx (D2,D1,n1)) + 2 = indx (D2,D1,j) by A311, XXREAL_0:1; ::_thesis: verum
end;
A312: 1 <= (indx (D2,D1,n1)) + 1 by NAT_1:12;
A313: indx (D2,D1,j) <= len H1(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 H1(D2) by INTEGRA1:def_7;
then A316: len (mid (H1(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 (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) <= (upper_bound (rng f)) * (vol (divset (D1,(n1 + 1))))
proof
percases ( (indx (D2,D1,j)) - (indx (D2,D1,n1)) = 1 or (indx (D2,D1,j)) - (indx (D2,D1,n1)) = 2 ) by A317, A310;
supposeA319: (indx (D2,D1,j)) - (indx (D2,D1,n1)) = 1 ; ::_thesis: Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) <= (upper_bound (rng f)) * (vol (divset (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;
1 = ((indx (D2,D1,j)) - ((indx (D2,D1,n1)) + 1)) + 1 by A319;
then (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . 1 = H1(D2) . ((1 + ((indx (D2,D1,n1)) + 1)) - 1) by A312, A313, FINSEQ_6:122
.= H1(D2) . ((indx (D2,D1,n1)) + 1) ;
then A325: mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))) = <*(H1(D2) . ((indx (D2,D1,n1)) + 1))*> by A316, A319, FINSEQ_1:40;
H1(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 H1(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 Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) <= (upper_bound (rng f)) * (vol (divset (D1,(n1 + 1)))) by A325, FINSOP_1:11; ::_thesis: verum
end;
supposeA326: (indx (D2,D1,j)) - (indx (D2,D1,n1)) = 2 ; ::_thesis: Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) <= (upper_bound (rng f)) * (vol (divset (D1,(n1 + 1))))
(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 H1(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 H1(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: H1(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 (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . 2 = H1(D2) . ((2 + ((indx (D2,D1,n1)) + 1)) - 1) by A335, A336, A333, FINSEQ_6:122
.= H1(D2) . (((indx (D2,D1,n1)) + 0) + 2) ;
((indx (D2,D1,j)) - ((indx (D2,D1,n1)) + 1)) + 1 >= 1 by A326;
then (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . 1 = H1(D2) . ((1 + ((indx (D2,D1,n1)) + 1)) - 1) by A326, A335, A336, A333, FINSEQ_6:122
.= H1(D2) . ((indx (D2,D1,n1)) + 1) ;
then mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))) = <*(H1(D2) . ((indx (D2,D1,n1)) + 1)),(H1(D2) . ((indx (D2,D1,n1)) + 2))*> by A316, A326, A339, FINSEQ_1:44;
then A340: Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) = (H1(D2) . ((indx (D2,D1,n1)) + 1)) + (H1(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;
H1(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 H1(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 (H1(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 Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) <= (upper_bound (rng f)) * (vol (divset (D1,(n1 + 1)))) by A282, A343, A329, A331; ::_thesis: verum
end;
end;
end;
A344: n1 + 1 <= len H1(D1) by A267, INTEGRA1:def_7;
then A345: len (mid (H1(D1),(n1 + 1),j)) = (j -' (n1 + 1)) + 1 by A244, A282, FINSEQ_6:118
.= (j - j) + 1 by A282, XREAL_1:233
.= 1 ;
(n1 + 1) + 1 <= j + 1 by A257, XREAL_1:6;
then 1 <= (j + 1) - (n1 + 1) by XREAL_1:19;
then (mid (H1(D1),(n1 + 1),j)) . 1 = H1(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 (H1(D1),(n1 + 1),j) = <*((lower_bound (rng (f | (divset (D1,(n1 + 1)))))) * (vol (divset (D1,(n1 + 1)))))*> by A345, FINSEQ_1:40;
then A346: Sum (mid (H1(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 (H1(D1),(n1 + 1),j)) >= (lower_bound (rng f)) * (vol (divset (D1,(n1 + 1)))) by A346, A347, XREAL_1:64;
then (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) - (Sum (mid (H1(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 (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) - (Sum (mid (H1(D1),(n1 + 1),j))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) by A350, XXREAL_0:2; ::_thesis: verum
end;
supposeA351: n1 + 1 < j ; ::_thesis: (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) - (Sum (mid (H1(D1),(n1 + 1),j))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
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 set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in B or x1 in A )
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 ; ::_thesis: x1 in A
then reconsider x1 = 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 x1 in A by A370, INTEGRA2:1; ::_thesis: verum
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
percases ( len MD1 = 1 or len MD1 <> 1 ) ;
supposeA375: len MD1 = 1 ; ::_thesis: ( lower_bound (divset (MD1,(len MD1))) = lower_bound (divset (D1,j)) & upper_bound (divset (MD1,(len MD1))) = upper_bound (divset (D1,j)) )
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 ( lower_bound (divset (MD1,(len MD1))) = lower_bound (divset (D1,j)) & upper_bound (divset (MD1,(len MD1))) = upper_bound (divset (D1,j)) ) by A265, A266, A354, A357, A361, A372, A375, A376, A377, FINSEQ_6:118, INTEGRA1:def_4; ::_thesis: verum
end;
supposeA378: len MD1 <> 1 ; ::_thesis: ( lower_bound (divset (MD1,(len MD1))) = lower_bound (divset (D1,j)) & upper_bound (divset (MD1,(len MD1))) = upper_bound (divset (D1,j)) )
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 ( lower_bound (divset (MD1,(len MD1))) = lower_bound (divset (D1,j)) & upper_bound (divset (MD1,(len MD1))) = upper_bound (divset (D1,j)) ) by A236, A245, A280, A381, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
A383: len MD1 in dom MD1 by FINSEQ_5:6;
A384: upper_bound (divset (MD1,(len MD1))) = MD1 . (len MD1)
proof
percases ( len MD1 = 1 or len MD1 <> 1 ) ;
suppose len MD1 = 1 ; ::_thesis: upper_bound (divset (MD1,(len MD1))) = MD1 . (len MD1)
hence upper_bound (divset (MD1,(len MD1))) = MD1 . (len MD1) by A383, INTEGRA1:def_4; ::_thesis: verum
end;
suppose len MD1 <> 1 ; ::_thesis: upper_bound (divset (MD1,(len MD1))) = MD1 . (len MD1)
hence upper_bound (divset (MD1,(len MD1))) = MD1 . (len MD1) by A383, INTEGRA1:def_4; ::_thesis: verum
end;
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 ; ::_thesis: contradiction
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__::_thesis:_contradiction
percases ( D2 . ((indx (D2,D1,n1)) + 1) in rng D1 or D2 . ((indx (D2,D1,n1)) + 1) in rng D ) by A14, A393, XBOOLE_0:def_3;
suppose D2 . ((indx (D2,D1,n1)) + 1) in rng D1 ; ::_thesis: contradiction
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; ::_thesis: verum
end;
supposeA397: D2 . ((indx (D2,D1,n1)) + 1) in rng D ; ::_thesis: contradiction
A398: D . i <= upper_bound (divset (D1,n1)) by A242, INTEGRA2:1;
A399: upper_bound (divset (D1,n1)) = D1 . n1
proof
percases ( n1 = 1 or n1 <> 1 ) ;
suppose n1 = 1 ; ::_thesis: upper_bound (divset (D1,n1)) = D1 . n1
hence upper_bound (divset (D1,n1)) = D1 . n1 by A241, INTEGRA1:def_4; ::_thesis: verum
end;
suppose n1 <> 1 ; ::_thesis: upper_bound (divset (D1,n1)) = D1 . n1
hence upper_bound (divset (D1,n1)) = D1 . n1 by A241, INTEGRA1:def_4; ::_thesis: verum
end;
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; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A407: j <= len H1(D1) by A266, INTEGRA1:def_7;
A408: for k being Nat st 1 <= k & k <= len (lower_volume (g,MD1)) holds
(lower_volume (g,MD1)) . k = (mid (H1(D1),(n1 + 1),j)) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len (lower_volume (g,MD1)) implies (lower_volume (g,MD1)) . k = (mid (H1(D1),(n1 + 1),j)) . k )
assume that
A409: 1 <= k and
A410: k <= len (lower_volume (g,MD1)) ; ::_thesis: (lower_volume (g,MD1)) . k = (mid (H1(D1),(n1 + 1),j)) . k
A411: k in Seg (len (lower_volume (g,MD1))) by A409, A410, FINSEQ_1:1;
then A412: k in Seg (len MD1) by INTEGRA1:def_7;
then A413: k in dom MD1 by FINSEQ_1:def_3;
k in dom MD1 by A412, FINSEQ_1:def_3;
then A414: (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
A415: n1 + 1 = 1 + k2 ;
A416: 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, A415, XXREAL_0:2;
then A417: k + k2 in Seg (len D1) by A416, FINSEQ_1:1;
then A418: k + k2 in dom D1 by FINSEQ_1:def_3;
1 + 1 <= k + k2 by A258, A409, A415, XREAL_1:7;
then A419: 1 < k + k2 by NAT_1:13;
A420: k2 = (n1 + 1) - 1 by A415;
A421: ( lower_bound (divset (D1,(k + k2))) = lower_bound (divset (MD1,k)) & upper_bound (divset (D1,(k + k2))) = upper_bound (divset (MD1,k)) )
proof
percases ( k = 1 or k <> 1 ) ;
supposeA422: k = 1 ; ::_thesis: ( lower_bound (divset (D1,(k + k2))) = lower_bound (divset (MD1,k)) & upper_bound (divset (D1,(k + k2))) = upper_bound (divset (MD1,k)) )
then upper_bound (divset (MD1,k)) = MD1 . k by A413, INTEGRA1:def_4;
then A423: upper_bound (divset (MD1,k)) = D1 . ((k + (n1 + 1)) - 1) by A257, A266, A244, A357, A371, A409, A410, A411, FINSEQ_6:122;
lower_bound (divset (MD1,k)) = D1 . n1 by A354, A413, A422, INTEGRA1:def_4;
hence ( lower_bound (divset (D1,(k + k2))) = lower_bound (divset (MD1,k)) & upper_bound (divset (D1,(k + k2))) = upper_bound (divset (MD1,k)) ) by A420, A419, A418, A422, A423, INTEGRA1:def_4; ::_thesis: verum
end;
supposeA424: k <> 1 ; ::_thesis: ( lower_bound (divset (D1,(k + k2))) = lower_bound (divset (MD1,k)) & upper_bound (divset (D1,(k + k2))) = upper_bound (divset (MD1,k)) )
then upper_bound (divset (MD1,k)) = MD1 . k by A413, INTEGRA1:def_4;
then A425: upper_bound (divset (MD1,k)) = D1 . ((k + (n1 + 1)) - 1) by A257, A266, A244, A357, A371, A409, A410, A411, FINSEQ_6:122;
A426: k - 1 <= (j - (n1 + 1)) + 1 by A371, A410, XREAL_1:146, XXREAL_0:2;
A427: lower_bound (divset (MD1,k)) = MD1 . (k - 1) by A413, A424, INTEGRA1:def_4;
A428: k - 1 in dom MD1 by A413, A424, 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, A428, A426, A427, FINSEQ_6:122;
hence ( lower_bound (divset (D1,(k + k2))) = lower_bound (divset (MD1,k)) & upper_bound (divset (D1,(k + k2))) = upper_bound (divset (MD1,k)) ) by A415, A419, A418, A425, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
divset (MD1,k) = [.(lower_bound (divset (MD1,k))),(upper_bound (divset (MD1,k))).] by INTEGRA1:4;
then A429: divset (D1,(k + k2)) = divset (MD1,k) by A421, INTEGRA1:4;
A430: k + k2 in dom D1 by A417, FINSEQ_1:def_3;
A431: (mid (H1(D1),(n1 + 1),j)) . k = H1(D1) . ((k + (n1 + 1)) - 1) by A257, A244, A371, A407, A409, A410, A411, FINSEQ_6:122
.= (lower_bound (rng (f | (divset (D1,(k + k2)))))) * (vol (divset (D1,(k + k2)))) by A415, A430, INTEGRA1:def_7 ;
k in dom MD1 by A412, FINSEQ_1:def_3;
then divset (D1,(k + k2)) c= B by A429, INTEGRA1:8;
hence (lower_volume (g,MD1)) . k = (mid (H1(D1),(n1 + 1),j)) . k by A414, A431, A429, FUNCT_1:51; ::_thesis: verum
end;
A432: g | B is bounded
proof
consider a being real number such that
A433: for x being set st x in A /\ (dom f) holds
a <= f . x by A1, RFUNCT_1:71;
for x being set st x in B /\ (dom g) holds
a <= g . x
proof
let x be set ; ::_thesis: ( x in B /\ (dom g) implies a <= g . x )
A434: (dom f) /\ B c= (dom f) /\ A by A364, XBOOLE_1:26;
assume x in B /\ (dom g) ; ::_thesis: a <= g . x
then A435: x in dom g by XBOOLE_0:def_4;
then x in (dom f) /\ B by RELAT_1:61;
then a <= f . x by A433, A434;
hence a <= g . x by A435, FUNCT_1:47; ::_thesis: verum
end;
then A436: g | B is bounded_below by RFUNCT_1:71;
consider a being real number such that
A437: for x being set st x in A /\ (dom f) holds
f . x <= a by A1, RFUNCT_1:70;
for x being set st x in B /\ (dom g) holds
g . x <= a
proof
let x be set ; ::_thesis: ( x in B /\ (dom g) implies g . x <= a )
A438: (dom f) /\ B c= (dom f) /\ A by A364, XBOOLE_1:26;
assume x in B /\ (dom g) ; ::_thesis: g . x <= a
then A439: x in dom g by XBOOLE_0:def_4;
then x in (dom f) /\ B by RELAT_1:61;
then a >= f . x by A437, A438;
hence g . x <= a by A439, FUNCT_1:47; ::_thesis: verum
end;
then g | B is bounded_above by RFUNCT_1:70;
hence g | B is bounded by A436; ::_thesis: verum
end;
rng f is bounded_below by A1, INTEGRA1:11;
then A440: 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 A440, XREAL_1:13;
then A441: ((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;
A442: n1 < j - 1 by A351, XREAL_1:20;
A443: indx (D2,D1,j) <= len H1(D2) by A263, INTEGRA1:def_7;
A444: len MD2 = ((indx (D2,D1,j)) -' (indx (D2,D1,(n1 + 1)))) + 1 by A272, A270, A277, A262, A263, A358, FINSEQ_6:118;
then A445: len MD2 = ((indx (D2,D1,j)) - (indx (D2,D1,(n1 + 1)))) + 1 by A272, XREAL_1:233;
then A446: len (lower_volume (g,MD2)) = ((indx (D2,D1,j)) - ((indx (D2,D1,n1)) + 1)) + 1 by A390, INTEGRA1:def_7;
for x1 being set st x1 in (rng MD1) \/ {(D . (i + 1))} holds
x1 in rng MD2
proof
let x1 be set ; ::_thesis: ( x1 in (rng MD1) \/ {(D . (i + 1))} implies x1 in rng MD2 )
assume A447: x1 in (rng MD1) \/ {(D . (i + 1))} ; ::_thesis: x1 in rng MD2
then reconsider x1 = x1 as Real ;
now__::_thesis:_x1_in_rng_MD2
percases ( x1 in rng MD1 or x1 in {(D . (i + 1))} ) by A447, XBOOLE_0:def_3;
supposeA448: x1 in rng MD1 ; ::_thesis: x1 in rng MD2
rng MD1 <> {} ;
then 1 in dom MD1 by FINSEQ_3:32;
then A449: 1 <= len MD1 by FINSEQ_3:25;
rng MD1 c= rng D1 by A357, FINSEQ_6:119;
then A450: x1 in rng D1 by A448;
rng D1 c= rng D2 by A13, INTEGRA1:def_18;
then consider k being Element of NAT such that
A451: k in dom D2 and
A452: D2 . k = x1 by A450, 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, A448, Th16;
then A453: indx (D2,D1,(n1 + 1)) <= k by A269, A451, A452, SEQM_3:def_1;
then consider n being Nat such that
A454: k + 1 = (indx (D2,D1,(n1 + 1))) + n by NAT_1:10, NAT_1:12;
A455: 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, A449, A455, FINSEQ_6:122;
then x1 <= D2 . (indx (D2,D1,j)) by A238, A448, Th16;
then k <= indx (D2,D1,j) by A261, A451, A452, 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 A456: (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 A453, XREAL_1:6;
then A457: 1 <= (k + 1) - (indx (D2,D1,(n1 + 1))) by XREAL_1:19;
then A458: n in dom MD2 by A445, A456, A454, FINSEQ_3:25;
n in NAT by ORDINAL1:def_12;
then MD2 . n = D2 . ((n + (indx (D2,D1,(n1 + 1)))) - 1) by A272, A270, A263, A358, A457, A456, A454, FINSEQ_6:122
.= D2 . k by A454 ;
hence x1 in rng MD2 by A452, A458, FUNCT_1:def_3; ::_thesis: verum
end;
suppose x1 in {(D . (i + 1))} ; ::_thesis: x1 in rng MD2
then A459: x1 = D . (i + 1) by TARSKI:def_1;
reconsider j1 = j - 1 as Element of NAT by A236, A245, A280, INTEGRA1:7;
A460: 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 being Element of NAT such that
A461: k in dom D2 and
A462: x1 = D2 . k by A459, A460, PARTFUN1:3;
D . (i + 1) <= upper_bound (divset (D1,j)) by A237, INTEGRA2:1;
then x1 <= D1 . j by A236, A245, A280, A459, INTEGRA1:def_4;
then A463: D2 . k <= D2 . (indx (D2,D1,j)) by A13, A236, A462, INTEGRA1:def_19;
n1 < j1 by A351, XREAL_1:20;
then A464: n1 + 1 <= j1 by NAT_1:13;
j - 1 in dom D1 by A236, A245, A280, INTEGRA1:7;
then A465: D1 . (n1 + 1) <= D1 . (j - 1) by A268, A464, SEQ_4:137;
lower_bound (divset (D1,j)) <= D . (i + 1) by A237, INTEGRA2:1;
then D1 . (j - 1) <= x1 by A236, A245, A280, A459, INTEGRA1:def_4;
then D2 . (indx (D2,D1,(n1 + 1))) <= D2 . k by A271, A462, A465, XXREAL_0:2;
hence x1 in rng MD2 by A269, A261, A358, A461, A462, A463, Th17; ::_thesis: verum
end;
end;
end;
hence x1 in rng MD2 ; ::_thesis: verum
end;
then A466: (rng MD1) \/ {(D . (i + 1))} c= rng MD2 by TARSKI:def_3;
rng MD2 <> {} ;
then 1 in dom MD2 by FINSEQ_3:32;
then A467: 1 <= len MD2 by FINSEQ_3:25;
A468: ((len MD2) - 1) + (indx (D2,D1,(n1 + 1))) = indx (D2,D1,j) by A445;
for x1 being set st x1 in rng MD2 holds
x1 in (rng MD1) \/ {(D . (i + 1))}
proof
let x1 be set ; ::_thesis: ( x1 in rng MD2 implies x1 in (rng MD1) \/ {(D . (i + 1))} )
assume A469: x1 in rng MD2 ; ::_thesis: x1 in (rng MD1) \/ {(D . (i + 1))}
then reconsider x1 = x1 as Real ;
MD2 . 1 = D2 . (indx (D2,D1,(n1 + 1))) by A270, A277, A262, A263, A358, FINSEQ_6:118;
then A470: D1 . (n1 + 1) <= x1 by A271, A469, Th16;
MD2 . (len MD2) = D2 . (indx (D2,D1,j)) by A272, A270, A263, A358, A467, A444, A468, FINSEQ_6:122;
then A471: x1 <= D1 . j by A238, A469, Th16;
A472: rng MD2 c= rng D2 by A358, FINSEQ_6:119;
now__::_thesis:_x1_in_(rng_MD1)_\/_{(D_._(i_+_1))}
percases ( x1 in rng D1 or x1 in rng D ) by A14, A469, A472, XBOOLE_0:def_3;
suppose x1 in rng D1 ; ::_thesis: x1 in (rng MD1) \/ {(D . (i + 1))}
then consider k being Element of NAT such that
A473: k in dom D1 and
A474: D1 . k = x1 by PARTFUN1:3;
A475: n1 + 1 <= k by A268, A470, A473, A474, SEQM_3:def_1;
then A476: 1 <= k - n1 by XREAL_1:19;
n1 <= n1 + 1 by NAT_1:11;
then consider n being Nat such that
A477: k = n1 + n by A475, NAT_1:10, XXREAL_0:2;
A478: k <= j by A236, A471, A473, A474, SEQM_3:def_1;
then k - n1 <= len MD1 by A363, XREAL_1:9;
then n in dom MD1 by A476, A477, FINSEQ_3:25;
then A479: MD1 . n in rng MD1 by FUNCT_1:def_3;
(j - (n1 + 1)) + 1 = j - n1 ;
then A480: k - n1 <= (j - (n1 + 1)) + 1 by A478, XREAL_1:9;
n in NAT by ORDINAL1:def_12;
then MD1 . n = D1 . (((k - n1) - 1) + (n1 + 1)) by A257, A266, A244, A357, A476, A480, A477, FINSEQ_6:122
.= D1 . k ;
hence x1 in (rng MD1) \/ {(D . (i + 1))} by A474, A479, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose x1 in rng D ; ::_thesis: x1 in (rng MD1) \/ {(D . (i + 1))}
then consider n being Element of NAT such that
A481: n in dom D and
A482: D . n = x1 by PARTFUN1:3;
A483: not i + 1 < n
proof
A484: upper_bound (divset (D1,j)) = D1 . j
proof
percases ( j = 1 or j <> 1 ) ;
suppose j = 1 ; ::_thesis: upper_bound (divset (D1,j)) = D1 . j
hence upper_bound (divset (D1,j)) = D1 . j by A236, INTEGRA1:def_4; ::_thesis: verum
end;
suppose j <> 1 ; ::_thesis: upper_bound (divset (D1,j)) = D1 . j
hence upper_bound (divset (D1,j)) = D1 . j by A236, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
consider y1 being Real such that
A485: y1 = D . (i + 1) ;
A486: D . n in rng D by A481, FUNCT_1:def_3;
assume i + 1 < n ; ::_thesis: contradiction
then A487: D . (i + 1) < D . n by A235, A481, 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 A487, XXREAL_0:2;
then D . n in divset (D1,j) by A471, A482, A484, INTEGRA2:1;
then A488: x1 in (rng D) /\ (divset (D1,j)) by A482, A486, 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, A485, XBOOLE_0:def_4;
hence contradiction by A11, A236, A482, A487, A488, A485, Th5; ::_thesis: verum
end;
A489: upper_bound (divset (D1,n1)) = D1 . n1
proof
percases ( n1 = 1 or n1 <> 1 ) ;
suppose n1 = 1 ; ::_thesis: upper_bound (divset (D1,n1)) = D1 . n1
hence upper_bound (divset (D1,n1)) = D1 . n1 by A241, INTEGRA1:def_4; ::_thesis: verum
end;
suppose n1 <> 1 ; ::_thesis: upper_bound (divset (D1,n1)) = D1 . n1
hence upper_bound (divset (D1,n1)) = D1 . n1 by A241, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
D . i <= upper_bound (divset (D1,n1)) by A242, INTEGRA2:1;
then D . i < D1 . (n1 + 1) by A353, A489, XXREAL_0:2;
then D . i < D . n by A470, A482, XXREAL_0:2;
then i < n by A240, A481, 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 A482, A483, TARSKI:def_1;
hence x1 in (rng MD1) \/ {(D . (i + 1))} by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
hence x1 in (rng MD1) \/ {(D . (i + 1))} ; ::_thesis: verum
end;
then rng MD2 c= (rng MD1) \/ {(D . (i + 1))} by TARSKI:def_3;
then A490: rng MD2 = (rng MD1) \/ {(D . (i + 1))} by A466, XBOOLE_0:def_10;
delta MD1 in rng (upper_volume ((chi (B,B)),MD1)) by XXREAL_2:def_8;
then consider k being Element of NAT such that
A491: k in dom (upper_volume ((chi (B,B)),MD1)) and
A492: (upper_volume ((chi (B,B)),MD1)) . k = delta MD1 by PARTFUN1:3;
A493: k in Seg (len (upper_volume ((chi (B,B)),MD1))) by A491, FINSEQ_1:def_3;
then A494: k in Seg (len MD1) by INTEGRA1:def_6;
then A495: k in dom MD1 by FINSEQ_1:def_3;
A496: k <= len MD1 by A494, FINSEQ_1:1;
then k + n1 <= j by A363, XREAL_1:19;
then A497: k + n1 <= len D1 by A266, XXREAL_0:2;
A498: 1 <= k by A493, FINSEQ_1:1;
A499: n1 + 1 > 1 by A280, NAT_1:13;
then n1 > 1 - 1 by XREAL_1:19;
then A500: k < k + n1 by XREAL_1:29;
then 1 < k + n1 by A498, XXREAL_0:2;
then A501: k + n1 in dom D1 by A497, 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
percases ( k = 1 or k <> 1 ) ;
supposeA502: k = 1 ; ::_thesis: ( lower_bound (divset (MD1,k)) = lower_bound (divset (D1,(k + n1))) & upper_bound (divset (MD1,k)) = upper_bound (divset (D1,(k + n1))) )
then upper_bound (divset (MD1,k)) = MD1 . k by A495, INTEGRA1:def_4;
then A503: upper_bound (divset (MD1,k)) = D1 . ((k + (n1 + 1)) - 1) by A257, A266, A244, A357, A360, A498, A496, FINSEQ_6:122;
lower_bound (divset (D1,(k + n1))) = D1 . ((k + n1) - 1) by A498, A500, A501, INTEGRA1:def_4;
hence ( lower_bound (divset (MD1,k)) = lower_bound (divset (D1,(k + n1))) & upper_bound (divset (MD1,k)) = upper_bound (divset (D1,(k + n1))) ) by A354, A499, A495, A501, A502, A503, INTEGRA1:def_4; ::_thesis: verum
end;
supposeA504: k <> 1 ; ::_thesis: ( lower_bound (divset (MD1,k)) = lower_bound (divset (D1,(k + n1))) & upper_bound (divset (MD1,k)) = upper_bound (divset (D1,(k + n1))) )
then upper_bound (divset (MD1,k)) = MD1 . k by A495, INTEGRA1:def_4;
then A505: upper_bound (divset (MD1,k)) = D1 . ((k + (n1 + 1)) - 1) by A257, A266, A244, A357, A360, A498, A496, FINSEQ_6:122;
A506: lower_bound (divset (MD1,k)) = MD1 . (k - 1) by A495, A504, INTEGRA1:def_4;
A507: k - 1 in dom MD1 by A495, A504, INTEGRA1:7;
then A508: k - 1 <= len MD1 by FINSEQ_3:25;
1 <= k - 1 by A507, FINSEQ_3:25;
then lower_bound (divset (MD1,k)) = D1 . (((k - 1) + (n1 + 1)) - 1) by A257, A266, A244, A357, A360, A507, A508, A506, FINSEQ_6:122;
hence ( lower_bound (divset (MD1,k)) = lower_bound (divset (D1,(k + n1))) & upper_bound (divset (MD1,k)) = upper_bound (divset (D1,(k + n1))) ) by A498, A500, A501, A505, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
then divset (MD1,k) = [.(lower_bound (divset (D1,(k + n1)))),(upper_bound (divset (D1,(k + n1)))).] by INTEGRA1:4;
then A509: divset (MD1,k) = divset (D1,(k + n1)) by INTEGRA1:4;
k + n1 in Seg (len D1) by A501, FINSEQ_1:def_3;
then k + n1 in Seg (len (upper_volume ((chi (A,A)),D1))) by INTEGRA1:def_6;
then A510: k + n1 in dom (upper_volume ((chi (A,A)),D1)) by FINSEQ_1:def_3;
k in dom MD1 by A494, FINSEQ_1:def_3;
then delta MD1 = vol (divset (MD1,k)) by A492, INTEGRA1:20;
then delta MD1 = (upper_volume ((chi (A,A)),D1)) . (k + n1) by A501, A509, INTEGRA1:20;
then delta MD1 in rng (upper_volume ((chi (A,A)),D1)) by A510, FUNCT_1:def_3;
then delta MD1 <= max (rng (upper_volume ((chi (A,A)),D1))) by XXREAL_2:def_8;
then A511: delta MD1 <= delta D1 ;
(upper_bound (rng f)) - (lower_bound (rng f)) >= 0 by A1, Lm3, XREAL_1:48;
then A512: ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta MD1) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) by A511, XREAL_1:64;
lower_bound (divset (D1,j)) <= D . (i + 1) by A237, INTEGRA2:1;
then A513: D1 . (j - 1) <= D . (i + 1) by A236, A245, A280, INTEGRA1:def_4;
A514: D . (i + 1) <= upper_bound (divset (D1,j)) by A237, INTEGRA2:1;
A515: (indx (D2,D1,n1)) + 1 <= indx (D2,D1,j) by A272, A385, XXREAL_0:2;
A516: for k being Nat st 1 <= k & k <= len (lower_volume (g,MD2)) holds
(lower_volume (g,MD2)) . k = (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len (lower_volume (g,MD2)) implies (lower_volume (g,MD2)) . k = (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . k )
assume that
A517: 1 <= k and
A518: k <= len (lower_volume (g,MD2)) ; ::_thesis: (lower_volume (g,MD2)) . k = (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . k
A519: k in Seg (len (lower_volume (g,MD2))) by A517, A518, FINSEQ_1:1;
then A520: (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . k = H1(D2) . ((k + ((indx (D2,D1,n1)) + 1)) - 1) by A388, A446, A443, A515, A517, A518, FINSEQ_6:122;
1 <= (indx (D2,D1,n1)) + 1 by NAT_1:12;
then 1 + 1 <= k + ((indx (D2,D1,n1)) + 1) by A517, XREAL_1:7;
then A521: 1 <= (k + ((indx (D2,D1,n1)) + 1)) - 1 by XREAL_1:19;
consider k2 being Element of NAT such that
A522: (indx (D2,D1,n1)) + 1 = 1 + k2 ;
k <= (indx (D2,D1,j)) - (((indx (D2,D1,n1)) + 1) - 1) by A445, A390, A518, 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 H1(D2) by A443, XXREAL_0:2;
then k + k2 in Seg (len H1(D2)) by A521, A522, FINSEQ_1:1;
then A523: k + k2 in Seg (len D2) by INTEGRA1:def_7;
then k + k2 in dom D2 by FINSEQ_1:def_3;
then A524: (mid (H1(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 A520, A522, INTEGRA1:def_7;
A525: k in Seg (len MD2) by A519, INTEGRA1:def_7;
A526: ( 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, A517, A522, XREAL_1:7;
then A527: k + k2 > 1 by NAT_1:13;
A528: k in dom MD2 by A525, FINSEQ_1:def_3;
A529: k + k2 in dom D2 by A523, FINSEQ_1:def_3;
percases ( k = 1 or k <> 1 ) ;
supposeA530: k = 1 ; ::_thesis: ( lower_bound (divset (MD2,k)) = lower_bound (divset (D2,(k + k2))) & upper_bound (divset (MD2,k)) = upper_bound (divset (D2,(k + k2))) )
then upper_bound (divset (MD2,k)) = MD2 . k by A528, INTEGRA1:def_4;
then A531: upper_bound (divset (MD2,k)) = D2 . ((k + ((indx (D2,D1,n1)) + 1)) - 1) by A272, A263, A358, A388, A390, A446, A517, A518, A519, FINSEQ_6:122;
A532: lower_bound (divset (D2,(k + k2))) = D2 . ((k + k2) - 1) by A527, A529, INTEGRA1:def_4;
lower_bound (divset (MD2,k)) = D1 . n1 by A354, A528, A530, INTEGRA1:def_4;
hence ( lower_bound (divset (MD2,k)) = lower_bound (divset (D2,(k + k2))) & upper_bound (divset (MD2,k)) = upper_bound (divset (D2,(k + k2))) ) by A13, A241, A522, A527, A529, A530, A531, A532, INTEGRA1:def_4, INTEGRA1:def_19; ::_thesis: verum
end;
supposeA533: k <> 1 ; ::_thesis: ( lower_bound (divset (MD2,k)) = lower_bound (divset (D2,(k + k2))) & upper_bound (divset (MD2,k)) = upper_bound (divset (D2,(k + k2))) )
then upper_bound (divset (MD2,k)) = MD2 . k by A528, INTEGRA1:def_4;
then A534: upper_bound (divset (MD2,k)) = D2 . ((k + ((indx (D2,D1,n1)) + 1)) - 1) by A272, A263, A358, A388, A390, A446, A517, A518, A519, FINSEQ_6:122;
A535: k - 1 <= ((indx (D2,D1,j)) - ((indx (D2,D1,n1)) + 1)) + 1 by A446, A518, XREAL_1:146, XXREAL_0:2;
A536: lower_bound (divset (MD2,k)) = MD2 . (k - 1) by A528, A533, INTEGRA1:def_4;
A537: k - 1 in dom MD2 by A528, A533, 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, A537, A535, A536, FINSEQ_6:122;
hence ( lower_bound (divset (MD2,k)) = lower_bound (divset (D2,(k + k2))) & upper_bound (divset (MD2,k)) = upper_bound (divset (D2,(k + k2))) ) by A522, A527, A529, A534, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
divset (MD2,k) = [.(lower_bound (divset (MD2,k))),(upper_bound (divset (MD2,k))).] by INTEGRA1:4;
then A538: divset (MD2,k) = divset (D2,(k + k2)) by A526, INTEGRA1:4;
k in dom MD2 by A525, FINSEQ_1:def_3;
then divset (D2,(k + k2)) c= B by A538, INTEGRA1:8;
then A539: rng (f | (divset (D2,(k + k2)))) = rng (g | (divset (D2,(k + k2)))) by FUNCT_1:51;
k in dom MD2 by A525, FINSEQ_1:def_3;
hence (lower_volume (g,MD2)) . k = (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . k by A524, A538, A539, INTEGRA1:def_7; ::_thesis: verum
end;
lower_bound (divset (D1,j)) <= D . (i + 1) by A237, INTEGRA2:1;
then A540: D . (i + 1) in divset (MD1,(len MD1)) by A374, A514, INTEGRA2:1;
j - 1 in dom D1 by A236, A245, A280, INTEGRA1:7;
then D1 . n1 < D1 . (j - 1) by A241, A442, SEQM_3:def_1;
then D . (i + 1) > lower_bound B by A354, A513, 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, A432, A490, A540, A387, Th13;
then A541: (Sum (lower_volume (g,MD2))) - (Sum (lower_volume (g,MD1))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta MD1) by A441, XXREAL_0:2;
(indx (D2,D1,n1)) + 1 <= len H1(D2) by A386, INTEGRA1:def_7;
then len (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) = ((indx (D2,D1,j)) -' ((indx (D2,D1,n1)) + 1)) + 1 by A262, A388, A443, A515, FINSEQ_6:118;
then len (lower_volume (g,MD2)) = len (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) by A272, A385, A446, XREAL_1:233, XXREAL_0:2;
then A542: Sum (lower_volume (g,MD2)) = Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) by A516, FINSEQ_1:14;
n1 + 1 <= len H1(D1) by A267, INTEGRA1:def_7;
then len (mid (H1(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 (H1(D1),(n1 + 1),j)) by A371, A408, FINSEQ_1:14;
hence (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) - (Sum (mid (H1(D1),(n1 + 1),j))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) by A541, A512, A542, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) - (Sum (mid (H1(D1),(n1 + 1),j))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) ; ::_thesis: verum
end;
then A543: (H2(D2, indx (D2,D1,n1)) - H2(D1,n1)) + ((Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) - (Sum (mid (H1(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 A544: 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 H1(D2)) by INTEGRA1:def_7;
then indx (D2,D1,n1) in dom H1(D2) by FINSEQ_1:def_3;
then H2(D2, indx (D2,D1,n1)) = Sum (H1(D2) | (indx (D2,D1,n1))) by INTEGRA1:def_20
.= Sum (mid (H1(D2),1,(indx (D2,D1,n1)))) by A260, FINSEQ_6:116 ;
then H2(D2, indx (D2,D1,n1)) + (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) = Sum ((mid (H1(D2),1,(indx (D2,D1,n1)))) ^ (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) by RVSUM_1:75
.= Sum (mid (H1(D2),1,(indx (D2,D1,j)))) by A260, A544, A264, INTEGRA2:4
.= Sum (H1(D2) | (indx (D2,D1,j))) by A262, FINSEQ_6:116 ;
then H2(D2, indx (D2,D1,j)) = H2(D2, indx (D2,D1,n1)) + (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) by A279, INTEGRA1:def_20;
then (H2(D2, indx (D2,D1,n1)) - H2(D1,n1)) + ((Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) - (Sum (mid (H1(D1),(n1 + 1),j)))) = H2(D2, indx (D2,D1,j)) - H2(D1,j) by A278;
hence ex j being Element of NAT st
( j in dom D1 & D . (i + 1) in divset (D1,j) & H2(D2, indx (D2,D1,j)) - H2(D1,j) <= ((i + 1) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) ) by A236, A237, A543; ::_thesis: verum
end;
hence S1[i + 1] ; ::_thesis: verum
end;
for k being non empty Nat holds S1[k] from NAT_1:sch_10(A40, A231);
then S1[i] ;
hence ex j being Element of NAT st
( j in dom D1 & D . i in divset (D1,j) & H2(D2, indx (D2,D1,j)) - H2(D1,j) <= (i * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) ) by A17; ::_thesis: verum
end;
A545: 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 A546: 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 being Element of NAT such that
A547: j in dom D1 and
A548: D . (len D) in divset (D1,j) and
A549: H2(D2, indx (D2,D1,j)) - H2(D1,j) <= ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by A16;
A550: j = len D1
proof
assume A551: j <> len D1 ; ::_thesis: contradiction
j <= len D1 by A547, FINSEQ_3:25;
then j < len D1 by A551, XXREAL_0:1;
then D1 . j < D1 . (len D1) by A547, A545, SEQM_3:def_1;
then A552: D1 . j < upper_bound A by INTEGRA1:def_2;
A553: upper_bound (divset (D1,j)) < upper_bound A
proof
percases ( j = 1 or j <> 1 ) ;
suppose j = 1 ; ::_thesis: upper_bound (divset (D1,j)) < upper_bound A
hence upper_bound (divset (D1,j)) < upper_bound A by A547, A552, INTEGRA1:def_4; ::_thesis: verum
end;
suppose j <> 1 ; ::_thesis: upper_bound (divset (D1,j)) < upper_bound A
hence upper_bound (divset (D1,j)) < upper_bound A by A547, A552, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
D . (len D) <= upper_bound (divset (D1,j)) by A548, INTEGRA2:1;
hence contradiction by A553, INTEGRA1:def_2; ::_thesis: verum
end;
indx (D2,D1,(len D1)) in dom D2 by A13, A545, INTEGRA1:def_19;
then indx (D2,D1,(len D1)) = len D2 by A15, A546, SEQ_4:138;
then H2(D2, len D2) - (lower_sum (f,D1)) <= ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by A549, A550, INTEGRA1:43;
hence (lower_sum (f,D2)) - (lower_sum (f,D1)) <= ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by INTEGRA1:43; ::_thesis: verum
end;
hence ex D2 being Division of A 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 A12, A13, A14; ::_thesis: verum
end;
hence ex D2 being Division of A 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) ) ; ::_thesis: verum
end;
A554: lim (delta T) = 0 by A2, FDIFF_1:def_1;
A555: delta T is non-zero by A2;
A556: for e being Real st e > 0 holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
( 0 < (delta T) . m & (delta T) . m < e )
proof
let e be Real; ::_thesis: ( e > 0 implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
( 0 < (delta T) . m & (delta T) . m < e ) )
assume e > 0 ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
( 0 < (delta T) . m & (delta T) . m < e )
then consider n being Element of NAT such that
A557: for m being Element of NAT st n <= m holds
abs (((delta T) . m) - 0) < e by A4, A554, SEQ_2:def_7;
for m being Element of NAT st n <= m holds
( 0 < (delta T) . m & (delta T) . m < e )
proof
let m be Element of NAT ; ::_thesis: ( n <= m implies ( 0 < (delta T) . m & (delta T) . m < e ) )
A558: (delta T) . m = delta (T . m) by Def2;
delta (T . m) in rng (upper_volume ((chi (A,A)),(T . m))) by XXREAL_2:def_8;
then consider i being Element of NAT such that
A559: i in dom (upper_volume ((chi (A,A)),(T . m))) and
A560: delta (T . m) = (upper_volume ((chi (A,A)),(T . m))) . i by PARTFUN1:3;
consider D being Division of A such that
A561: D = T . m ;
i in Seg (len (upper_volume ((chi (A,A)),(T . m)))) by A559, FINSEQ_1:def_3;
then i in Seg (len D) by A561, INTEGRA1:def_6;
then i in dom D by FINSEQ_1:def_3;
then A562: delta (T . m) = vol (divset ((T . m),i)) by A560, A561, INTEGRA1:20;
assume n <= m ; ::_thesis: ( 0 < (delta T) . m & (delta T) . m < e )
then abs (((delta T) . m) - 0) < e by A557;
then A563: ((delta T) . m) + (abs (((delta T) . m) - 0)) < e + (abs (((delta T) . m) - 0)) by ABSVALUE:4, XREAL_1:8;
(delta T) . m <> 0 by A555, SEQ_1:5;
hence ( 0 < (delta T) . m & (delta T) . m < e ) by A563, A558, A562, INTEGRA1:9, XREAL_1:6; ::_thesis: verum
end;
hence ex n being Element of NAT st
for m being Element of NAT st n <= m holds
( 0 < (delta T) . m & (delta T) . m < e ) ; ::_thesis: verum
end;
A564: for e being real number st e > 0 holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((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 number ; ::_thesis: ( e > 0 implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((lower_sum (f,T)) . m) - (lower_integral f)) < e )
assume A565: e > 0 ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((lower_sum (f,T)) . m) - (lower_integral f)) < e
then A566: e / 2 > 0 by XREAL_1:139;
reconsider e = e as Real by XREAL_0:def_1;
A567: (upper_bound (rng f)) - (lower_bound (rng f)) >= 0 by A1, Lm3, XREAL_1:48;
A568: 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 being real number such that
A569: y in rng (lower_sum_set f) and
A570: (lower_integral f) - (e / 2) < y by A566, A568, SEQ_4:def_1;
consider D being Division of A such that
D in dom (lower_sum_set f) and
A571: y = (lower_sum_set f) . D and
A572: D . 1 > lower_bound A by A3, A569, Lm7;
deffunc H1( Nat) -> Element of REAL = vol (divset (D,$1));
set p = len D;
consider v being FinSequence of REAL such that
A573: ( len v = len D & ( for j being Nat st j in dom v holds
v . j = H1(j) ) ) from FINSEQ_2:sch_1();
consider v1 being non-decreasing FinSequence of REAL such that
A574: v,v1 are_fiberwise_equipotent by INTEGRA2:3;
defpred S1[ Nat] means ( $1 in dom v1 & v1 . $1 > 0 );
A575: dom v = Seg (len D) by A573, FINSEQ_1:def_3;
A576: ex k being Nat st S1[k]
proof
consider H being Function such that
dom H = dom v and
rng H = dom v1 and
H is one-to-one and
A577: v = v1 * H by A574, CLASSES1:77;
consider k being Element of NAT such that
A578: k in dom D and
A579: vol (divset (D,k)) > 0 by A3, Th2;
A580: dom D = Seg (len v) by A573, FINSEQ_1:def_3;
then H . k in dom v1 by A573, A575, A577, A578, FUNCT_1:11;
then reconsider Hk = H . k as Nat ;
v . k > 0 by A573, A575, A578, A579, A580;
then S1[Hk] by A573, A575, A577, A578, A580, FUNCT_1:11, FUNCT_1:12;
hence ex k being Nat st S1[k] ; ::_thesis: verum
end;
consider k being Nat such that
A581: ( S1[k] & ( for n being Nat st S1[n] holds
k <= n ) ) from NAT_1:sch_5(A576);
A582: 2 * (len D) > 0 by XREAL_1:129;
then A583: (2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1) > 0 by A567, XREAL_1:129;
min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) > 0
proof
percases ( min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) = v1 . k or min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) = e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) ) by XXREAL_0:15;
suppose min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) = v1 . k ; ::_thesis: min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) > 0
hence min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) > 0 by A581; ::_thesis: verum
end;
suppose min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) = e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) ; ::_thesis: min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) > 0
hence min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) > 0 by A565, A583, XREAL_1:139; ::_thesis: verum
end;
end;
end;
then consider n being Element of NAT such that
A584: for m being Element of NAT st n <= m holds
( 0 < (delta T) . m & (delta T) . m < min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) ) by A556;
take n ; ::_thesis: for m being Element of NAT st n <= m holds
abs (((lower_sum (f,T)) . m) - (lower_integral f)) < e
A585: y = lower_sum (f,D) by A571, INTEGRA1:def_11;
for m being Element of NAT st n <= m holds
abs (((lower_sum (f,T)) . m) - (lower_integral f)) < e
proof
A586: v1 . 1 > 0
proof
A587: for n1 being Element of NAT st n1 in dom D holds
vol (divset (D,n1)) > 0
proof
let n1 be Element of NAT ; ::_thesis: ( n1 in dom D implies vol (divset (D,n1)) > 0 )
assume A588: n1 in dom D ; ::_thesis: vol (divset (D,n1)) > 0
then A589: 1 <= n1 by FINSEQ_3:25;
percases ( n1 = 1 or n1 > 1 ) by A589, XXREAL_0:1;
supposeA590: n1 = 1 ; ::_thesis: vol (divset (D,n1)) > 0
then A591: upper_bound (divset (D,n1)) = D . n1 by A588, INTEGRA1:def_4;
lower_bound (divset (D,n1)) = lower_bound A by A588, A590, INTEGRA1:def_4;
then vol (divset (D,n1)) = (D . n1) - (lower_bound A) by A591, INTEGRA1:def_5;
hence vol (divset (D,n1)) > 0 by A572, A590, XREAL_1:50; ::_thesis: verum
end;
supposeA592: n1 > 1 ; ::_thesis: vol (divset (D,n1)) > 0
then A593: upper_bound (divset (D,n1)) = D . n1 by A588, INTEGRA1:def_4;
lower_bound (divset (D,n1)) = D . (n1 - 1) by A588, A592, INTEGRA1:def_4;
then A594: vol (divset (D,n1)) = (D . n1) - (D . (n1 - 1)) by A593, INTEGRA1:def_5;
n1 < n1 + 1 by XREAL_1:29;
then A595: n1 - 1 < n1 by XREAL_1:19;
n1 - 1 in dom D by A588, A592, INTEGRA1:7;
then D . (n1 - 1) < D . n1 by A588, A595, SEQM_3:def_1;
hence vol (divset (D,n1)) > 0 by A594, XREAL_1:50; ::_thesis: verum
end;
end;
end;
A596: k <= len v1 by A581, FINSEQ_3:25;
1 <= k by A581, FINSEQ_3:25;
then 1 <= len v1 by A596, XXREAL_0:2;
then 1 in dom v1 by FINSEQ_3:25;
then A597: v1 . 1 in rng v1 by FUNCT_1:def_3;
rng v = rng v1 by A574, CLASSES1:75;
then consider n1 being Element of NAT such that
A598: n1 in dom v and
A599: v1 . 1 = v . n1 by A597, PARTFUN1:3;
n1 in Seg (len D) by A573, A598, FINSEQ_1:def_3;
then A600: n1 in dom D by FINSEQ_1:def_3;
v1 . 1 = vol (divset (D,n1)) by A573, A598, A599;
hence v1 . 1 > 0 by A587, A600; ::_thesis: verum
end;
A601: v1 . k = min (rng (upper_volume ((chi (A,A)),D)))
proof
A602: k = 1
proof
len v1 = len v by A574, RFINSEQ:3;
then k in Seg (len v) by A581, FINSEQ_1:def_3;
then A603: 1 <= k by FINSEQ_1:1;
k in Seg (len v1) by A581, FINSEQ_1:def_3;
then k <= len v1 by FINSEQ_1:1;
then 1 <= len v1 by A603, XXREAL_0:2;
then A604: 1 in dom v1 by FINSEQ_3:25;
assume k <> 1 ; ::_thesis: contradiction
then k > 1 by A603, XXREAL_0:1;
hence contradiction by A581, A586, A604; ::_thesis: verum
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 being Element of NAT such that
A605: m in dom (upper_volume ((chi (A,A)),D)) and
A606: 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 A605, FINSEQ_1:def_3;
then A607: m in Seg (len D) by INTEGRA1:def_6;
then m in dom D by FINSEQ_1:def_3;
then min (rng (upper_volume ((chi (A,A)),D))) = vol (divset (D,m)) by A606, INTEGRA1:20;
then A608: v . m = min (rng (upper_volume ((chi (A,A)),D))) by A573, A575, A607;
A609: rng v = rng v1 by A574, CLASSES1:75;
m in dom v by A573, A607, FINSEQ_1:def_3;
then min (rng (upper_volume ((chi (A,A)),D))) in rng v by A608, FUNCT_1:def_3;
then consider m1 being Element of NAT such that
A610: m1 in dom v1 and
A611: min (rng (upper_volume ((chi (A,A)),D))) = v1 . m1 by A609, PARTFUN1:3;
v1 . k in rng v1 by A581, FUNCT_1:def_3;
then consider k2 being Element of NAT such that
A612: k2 in dom v and
A613: v1 . k = v . k2 by A609, PARTFUN1:3;
A614: k2 in Seg (len D) by A573, A612, FINSEQ_1:def_3;
then A615: k2 in dom D by FINSEQ_1:def_3;
k2 in Seg (len (upper_volume ((chi (A,A)),D))) by A614, INTEGRA1:def_6;
then A616: k2 in dom (upper_volume ((chi (A,A)),D)) by FINSEQ_1:def_3;
v1 . k = vol (divset (D,k2)) by A573, A612, A613;
then v1 . k = (upper_volume ((chi (A,A)),D)) . k2 by A615, INTEGRA1:20;
then v1 . k in rng (upper_volume ((chi (A,A)),D)) by A616, FUNCT_1:def_3;
then A617: v1 . k >= min (rng (upper_volume ((chi (A,A)),D))) by XXREAL_2:def_7;
m1 >= 1 by A610, FINSEQ_3:25;
then v1 . 1 <= min (rng (upper_volume ((chi (A,A)),D))) by A581, A602, A610, A611, INTEGRA2:2;
hence v1 . k = min (rng (upper_volume ((chi (A,A)),D))) by A602, A617, XXREAL_0:1; ::_thesis: verum
end;
(upper_bound (rng f)) - (lower_bound (rng f)) <= ((upper_bound (rng f)) - (lower_bound (rng f))) + 1 by XREAL_1:29;
then A618: (len D) * ((upper_bound (rng f)) - (lower_bound (rng f))) <= (len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1) by XREAL_1:64;
set sD = lower_sum (f,D);
set s = lower_integral f;
let m be Element of NAT ; ::_thesis: ( n <= m implies abs (((lower_sum (f,T)) . m) - (lower_integral f)) < e )
reconsider D1 = T . m as Division of A ;
A619: min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) <= e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) by XXREAL_0:17;
assume A620: n <= m ; ::_thesis: abs (((lower_sum (f,T)) . m) - (lower_integral f)) < e
then (delta T) . m < min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) by A584;
then A621: delta D1 < min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) by Def2;
(delta T) . m < min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) by A584, A620;
then (delta T) . m < e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) by A619, XXREAL_0:2;
then ((delta T) . m) * ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) < e by A582, A567, XREAL_1:79, XREAL_1:129;
then (((delta T) . m) * ((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1))) * 2 < e ;
then A622: ((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) * ((delta T) . m) < e / 2 by XREAL_1:81;
T . m in divs A by INTEGRA1:def_3;
then A623: T . m in dom (lower_sum_set f) by FUNCT_2:def_1;
(lower_sum (f,T)) . m = lower_sum (f,(T . m)) 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 A623, FUNCT_1:def_3;
then upper_bound (rng (lower_sum_set f)) >= (lower_sum (f,T)) . m by A568, SEQ_4:def_1;
then lower_integral f >= (lower_sum (f,T)) . m by INTEGRA1:def_15;
then A624: (lower_integral f) - ((lower_sum (f,T)) . m) >= 0 by XREAL_1:48;
0 < (delta T) . m by A584, A620;
then A625: ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * ((delta T) . m) <= ((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) * ((delta T) . m) by A618, XREAL_1:64;
set sD1 = lower_sum (f,(T . m));
consider D2 being Division of A such that
A626: D <= D2 and
D1 <= D2 and
A627: 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);
A628: ((lower_sum (f,D)) - (lower_sum (f,(T . m)))) - ((lower_sum (f,D2)) - (lower_sum (f,(T . m)))) = (lower_sum (f,D)) - (lower_sum (f,D2)) ;
min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) <= v1 . k by XXREAL_0:17;
then delta D1 < v1 . k by A621, 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, A601;
then A629: (lower_sum (f,D2)) - (lower_sum (f,D1)) <= ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by A627, Th6;
(lower_sum (f,D)) - (lower_sum (f,D2)) <= 0 by A1, A626, INTEGRA1:46, XREAL_1:47;
then A630: (lower_sum (f,D)) - (lower_sum (f,(T . m))) <= (lower_sum (f,D2)) - (lower_sum (f,(T . m))) by A628, XREAL_1:50;
delta D1 = (delta T) . m by Def2;
then (lower_sum (f,D2)) - (lower_sum (f,(T . m))) <= ((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) * ((delta T) . m) by A629, A625, XXREAL_0:2;
then (lower_sum (f,D)) - (lower_sum (f,(T . m))) <= ((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) * ((delta T) . m) by A630, XXREAL_0:2;
then (lower_sum (f,D)) - (lower_sum (f,(T . m))) < e / 2 by A622, XXREAL_0:2;
then A631: ((lower_sum (f,D)) - (lower_sum (f,(T . m)))) + (e / 2) < (e / 2) + (e / 2) by XREAL_1:6;
((lower_integral f) - (lower_sum (f,(T . m)))) + (lower_sum (f,(T . m))) < (lower_sum (f,D)) + (e / 2) by A570, A585, XREAL_1:19;
then (lower_integral f) - (lower_sum (f,(T . m))) < ((lower_sum (f,D)) + (e / 2)) - (lower_sum (f,(T . m))) by XREAL_1:20;
then (lower_integral f) - (lower_sum (f,(T . m))) < e by A631, XXREAL_0:2;
then (lower_integral f) - ((lower_sum (f,T)) . m) < e by INTEGRA2:def_3;
then abs ((lower_integral f) - ((lower_sum (f,T)) . m)) < e by A624, ABSVALUE:def_1;
then abs (- ((lower_integral f) - ((lower_sum (f,T)) . m))) < e by COMPLEX1:52;
hence abs (((lower_sum (f,T)) . m) - (lower_integral f)) < e ; ::_thesis: verum
end;
hence for m being Element of NAT st n <= m holds
abs (((lower_sum (f,T)) . m) - (lower_integral f)) < e ; ::_thesis: verum
end;
hence lower_sum (f,T) is convergent by SEQ_2:def_6; ::_thesis: lim (lower_sum (f,T)) = lower_integral f
hence lim (lower_sum (f,T)) = lower_integral f by A564, SEQ_2:def_7; ::_thesis: verum
end;
theorem :: INTEGRA3:21
for A being non empty closed_interval Subset of REAL
for f being Function of A,REAL
for T being DivSequence of A st f | A is bounded & delta T is 0 -convergent & delta T is non-zero & vol A <> 0 holds
( upper_sum (f,T) is convergent & lim (upper_sum (f,T)) = upper_integral f )
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being Function of A,REAL
for T being DivSequence of A st f | A is bounded & delta T is 0 -convergent & delta T is non-zero & vol A <> 0 holds
( upper_sum (f,T) is convergent & lim (upper_sum (f,T)) = upper_integral f )
let f be Function of A,REAL; ::_thesis: for T being DivSequence of A st f | A is bounded & delta T is 0 -convergent & delta T is non-zero & vol A <> 0 holds
( upper_sum (f,T) is convergent & lim (upper_sum (f,T)) = upper_integral f )
let T be DivSequence of A; ::_thesis: ( f | A is bounded & delta T is 0 -convergent & delta T is non-zero & vol A <> 0 implies ( upper_sum (f,T) is convergent & lim (upper_sum (f,T)) = upper_integral f ) )
assume A1: f | A is bounded ; ::_thesis: ( not delta T is 0 -convergent or not delta T is non-zero or not vol A <> 0 or ( upper_sum (f,T) is convergent & lim (upper_sum (f,T)) = upper_integral f ) )
A2: for D, D1 being Division of A ex D2 being Division of A 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
let D, D1 be Division of A; ::_thesis: ex D2 being Division of A 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)) )
consider D2 being Division of A 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 ex D2 being Division of A 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 A3, A4, A5, A6; ::_thesis: verum
end;
A7: for D, D1 being Division of A st delta D1 < min (rng (upper_volume ((chi (A,A)),D))) holds
ex D2 being Division of A 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
let D, D1 be Division of A; ::_thesis: ( delta D1 < min (rng (upper_volume ((chi (A,A)),D))) implies ex D2 being Division of A 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) ) )
assume A8: delta D1 < min (rng (upper_volume ((chi (A,A)),D))) ; ::_thesis: ex D2 being Division of A 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) )
ex D2 being Division of A 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 being 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 H1( Division of A) -> FinSequence of REAL = upper_volume (f,$1);
deffunc H2( Division of A, Nat) -> Element of REAL = (PartSums (upper_volume (f,$1))) . $2;
A12: len D2 in dom D2 by FINSEQ_5:6;
A13: for i being Element of NAT st i in dom D holds
ex j being Element of NAT st
( j in dom D1 & D . i in divset (D1,j) & H2(D1,j) - H2(D2, indx (D2,D1,j)) <= (i * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) )
proof
defpred S1[ non empty Nat] means ( $1 in dom D implies ex j being Element of NAT st
( j in dom D1 & D . $1 in divset (D1,j) & H2(D1,j) - H2(D2, indx (D2,D1,j)) <= ($1 * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) ) );
let i be Element of NAT ; ::_thesis: ( i in dom D implies ex j being Element of NAT st
( j in dom D1 & D . i in divset (D1,j) & H2(D1,j) - H2(D2, indx (D2,D1,j)) <= (i * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) ) )
assume A14: i in dom D ; ::_thesis: ex j being Element of NAT st
( j in dom D1 & D . i in divset (D1,j) & H2(D1,j) - H2(D2, indx (D2,D1,j)) <= (i * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) )
then A15: i in Seg (len D) by FINSEQ_1:def_3;
A16: for i, j being Element of NAT st i in dom D & j in dom D1 & D . i in divset (D1,j) holds
j >= 2
proof
let i, j be Element of NAT ; ::_thesis: ( i in dom D & j in dom D1 & D . i in divset (D1,j) implies j >= 2 )
assume A17: i in dom D ; ::_thesis: ( not j in dom D1 or not D . i in divset (D1,j) or j >= 2 )
assume that
A18: j in dom D1 and
A19: D . i in divset (D1,j) ; ::_thesis: j >= 2
assume j < 2 ; ::_thesis: contradiction
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
percases ( i = 1 or i <> 1 ) ;
supposeA23: i = 1 ; ::_thesis: delta D1 >= min (rng (upper_volume ((chi (A,A)),D)))
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 delta D1 >= min (rng (upper_volume ((chi (A,A)),D))) by A27, XXREAL_0:2; ::_thesis: verum
end;
supposeA30: i <> 1 ; ::_thesis: delta D1 >= min (rng (upper_volume ((chi (A,A)),D)))
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 delta D1 >= min (rng (upper_volume ((chi (A,A)),D))) by A35, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence contradiction by A8; ::_thesis: verum
end;
A36: S1[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 being Element of NAT such that
A38: j in dom D1 and
A39: D . 1 in divset (D1,j) by Th3, INTEGRA1:6;
H2(D1,j) - H2(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 ; ::_thesis: contradiction
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 ) ; ::_thesis: contradiction
percases ( D2 . ID1 in rng D1 or D2 . ID2 in rng D1 ) by A66;
suppose D2 . ID1 in rng D1 ; ::_thesis: contradiction
then consider n being Element of NAT 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; ::_thesis: verum
end;
suppose D2 . ID2 in rng D1 ; ::_thesis: contradiction
then consider n being Element of NAT 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; ::_thesis: verum
end;
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; ::_thesis: verum
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;
(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) = <*((upper_volume (f,D1)) . j)*> 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__::_thesis:_(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)
percases ( len (mid ((upper_volume (f,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) = 1 or len (mid ((upper_volume (f,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) = 2 ) by A90, A89, Lm2;
supposeA91: len (mid ((upper_volume (f,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) = 1 ; ::_thesis: (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)
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;
(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))) = <*((upper_volume (f,D2)) . ((indx (D2,D1,j1)) + 1))*> 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 (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) by A94, A95; ::_thesis: verum
end;
supposeA100: len (mid ((upper_volume (f,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) = 2 ; ::_thesis: (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)
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 = H1(D2) . ((2 + ((indx (D2,D1,j1)) + 1)) -' 1) by A49, A84, A85, A78, A86, A100, FINSEQ_6:118
.= H1(D2) . ((2 + ((indx (D2,D1,j1)) + 1)) - 1) by A102, XREAL_1:233
.= H1(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 (H1(D1),j,j))) - (Sum (mid (H1(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,((indx (D2,D1,j1)) + 1)))));
set VR = vol (divset (D2,((indx (D2,D1,j1)) + 1)));
set B = vol (divset (D2,((indx (D2,D1,j1)) + 1)));
set C = vol (divset (D2,((indx (D2,D1,j1)) + 2)));
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;
(indx (D2,D1,j1)) + 1 in dom D2 by A87, FINSEQ_1:def_3;
then divset (D2,((indx (D2,D1,j1)) + 1)) c= A by INTEGRA1:8;
then upper_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 1))))) >= lower_bound (rng f) by A1, Lm4;
then A120: (upper_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) >= (lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) 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,((indx (D2,D1,j1)) + 2)) c= A by A121, INTEGRA1:8;
then A123: upper_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 2))))) >= lower_bound (rng f) by A1, Lm4;
reconsider A = upper_bound (rng (f | (divset (D1,j)))) as real number ;
A124: ((upper_volume (f,D1)) . j) - (A * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) = A * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) by A117;
(upper_bound (rng (f | (divset (D1,j))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) <= (upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) by A106, A119, XREAL_1:64;
then Sum (mid (H1(D1),j,j)) <= ((upper_bound (rng (f | (divset (D1,j))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) + ((upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) by A82, A124, XREAL_1:20;
then A125: (Sum (mid (H1(D1),j,j))) - ((upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) <= (upper_bound (rng (f | (divset (D1,j))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by XREAL_1:20;
(upper_bound (rng (f | (divset (D1,j))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) <= (upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A104, A119, XREAL_1:64;
then (Sum (mid (H1(D1),j,j))) - ((upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) <= (upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A125, XXREAL_0:2;
then A126: Sum (mid (H1(D1),j,j)) <= ((upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) by XREAL_1:20;
Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) = ((upper_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 2)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + (H1(D2) . ((indx (D2,D1,j1)) + 1)) by A103, A122, A121, INTEGRA1:def_6
.= ((upper_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 2)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((upper_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) by A88, INTEGRA1:def_6 ;
then (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) - ((upper_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) >= (lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2)))) by A106, A123, XREAL_1:64;
then Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) >= ((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((upper_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) by XREAL_1:19;
then (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) - ((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) >= (upper_bound (rng (f | (divset (D2,((indx (D2,D1,j1)) + 1)))))) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by XREAL_1:19;
then (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) - ((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) >= (lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))) by A120, XXREAL_0:2;
then Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j)))) >= ((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1))))) by XREAL_1:19;
then (Sum (mid (H1(D1),j,j))) - (Sum (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) <= (((upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((upper_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))))) - (((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 2))))) + ((lower_bound (rng f)) * (vol (divset (D2,((indx (D2,D1,j1)) + 1)))))) by A126, XREAL_1:13;
hence (Sum (mid (H1(D1),j,j))) - (Sum (mid (H1(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))))) ; ::_thesis: verum
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 (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) by A116, A118, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (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) ; ::_thesis: verum
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 set st x1 in rng (D1 | j1) holds
x1 in rng (D2 | (indx (D2,D1,j1)))
proof
let x1 be set ; ::_thesis: ( x1 in rng (D1 | j1) implies x1 in rng (D2 | (indx (D2,D1,j1))) )
assume x1 in rng (D1 | j1) ; ::_thesis: x1 in rng (D2 | (indx (D2,D1,j1)))
then consider k being Element of NAT 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 being Element of NAT 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 x1 in rng (D2 | (indx (D2,D1,j1))) by A141, A144, FUNCT_1:def_3; ::_thesis: verum
end;
then A145: rng (D1 | j1) c= rng (D2 | (indx (D2,D1,j1))) by TARSKI:def_3;
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 set st x1 in rng (D2 | (indx (D2,D1,j1))) holds
x1 in rng (D1 | j1)
proof
let x1 be set ; ::_thesis: ( x1 in rng (D2 | (indx (D2,D1,j1))) implies x1 in rng (D1 | j1) )
assume x1 in rng (D2 | (indx (D2,D1,j1))) ; ::_thesis: x1 in rng (D1 | j1)
then consider k being Element of NAT 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 ; ::_thesis: D2 . k in rng (D1 | j1)
then consider m being Element of NAT 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 D2 . k in rng (D1 | j1) by A159, FUNCT_1:def_3; ::_thesis: verum
end;
A160: ( D2 . k in rng D implies D2 . k = D1 . j1 )
proof
assume D2 . k in rng D ; ::_thesis: D2 . k = D1 . j1
then consider n being Element of NAT 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 D2 . k = D1 . j1 by A153, XXREAL_0:1; ::_thesis: verum
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 ; ::_thesis: D2 . k in rng (D1 | j1)
j1 in Seg j1 by A146, FINSEQ_1:1;
hence D2 . k in rng (D1 | j1) by A41, A160, A165, A164, RFINSEQ:6; ::_thesis: verum
end;
D2 . k in rng D2 by A151, FUNCT_1:def_3;
hence x1 in rng (D1 | j1) by A11, A44, A149, A150, A163, A154, RFINSEQ:6, XBOOLE_0:def_3; ::_thesis: verum
end;
then rng (D2 | (indx (D2,D1,j1))) c= rng (D1 | j1) by TARSKI:def_3;
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, INTEGRA1:35;
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 being Element of NAT st 1 <= k & k <= j1 holds
k = indx (D2,D1,k)
proof
let k be Element of NAT ; ::_thesis: ( 1 <= k & k <= j1 implies k = indx (D2,D1,k) )
assume that
A170: 1 <= k and
A171: k <= j1 ; ::_thesis: k = indx (D2,D1,k)
assume A172: k <> indx (D2,D1,k) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( k > indx (D2,D1,k) or k < indx (D2,D1,k) ) by A172, XXREAL_0:1;
supposeA173: k > indx (D2,D1,k) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
supposeA180: k < indx (D2,D1,k) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A187: for k being 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; ::_thesis: ( 1 <= k & k <= len ((upper_volume (f,D1)) | j1) implies ((upper_volume (f,D1)) | j1) . k = ((upper_volume (f,D2)) | (indx (D2,D1,j1))) . k )
assume that
A189: 1 <= k and
A190: k <= len ((upper_volume (f,D1)) | j1) ; ::_thesis: ((upper_volume (f,D1)) | j1) . k = ((upper_volume (f,D2)) | (indx (D2,D1,j1))) . k
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
percases ( k = 1 or k <> 1 ) ;
supposeA201: k = 1 ; ::_thesis: ( 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)))) )
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 ( 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)))) ) by A198, A199, A203, A202, INTEGRA1:def_4; ::_thesis: verum
end;
supposeA204: k <> 1 ; ::_thesis: ( 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)))) )
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 ( 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)))) ) by A10, A197, A211, A206, A209, A210, A207, INTEGRA1:def_19; ::_thesis: verum
end;
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 ((upper_volume (f,D1)) | j1) . k = ((upper_volume (f,D2)) | (indx (D2,D1,j1))) . k by A214; ::_thesis: verum
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 H1(D2)) by A128, INTEGRA1:def_6;
then A221: indx (D2,D1,j) in dom H1(D2) by FINSEQ_1:def_3;
indx (D2,D1,j) <= len D2 by A128, FINSEQ_1:1;
then A222: indx (D2,D1,j) <= len H1(D2) by INTEGRA1:def_6;
j in Seg (len H1(D1)) by A219, INTEGRA1:def_6;
then A223: j in dom H1(D1) by FINSEQ_1:def_3;
j <= len D1 by A219, FINSEQ_1:1;
then A224: j <= len H1(D1) by INTEGRA1:def_6;
j1 in Seg (len D1) by A41, FINSEQ_1:def_3;
then j1 in Seg (len H1(D1)) by INTEGRA1:def_6;
then j1 in dom H1(D1) by FINSEQ_1:def_3;
then H2(D1,j1) = Sum (H1(D1) | j1) by INTEGRA1:def_20;
then H2(D1,j1) + (Sum (mid (H1(D1),j,j))) = Sum ((H1(D1) | j1) ^ (mid (H1(D1),j,j))) by RVSUM_1:75
.= Sum ((mid (H1(D1),1,j1)) ^ (mid (H1(D1),(j1 + 1),j))) by A146, FINSEQ_6:116
.= Sum (mid (H1(D1),1,j)) by A146, A224, A127, INTEGRA2:4
.= Sum (H1(D1) | j) by A220, FINSEQ_6:116 ;
then A225: H2(D1,j1) + (Sum (mid ((upper_volume (f,D1)),j,j))) = H2(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 H1(D2)) by INTEGRA1:def_6;
then indx (D2,D1,j1) in dom H1(D2) by FINSEQ_1:def_3;
then H2(D2, indx (D2,D1,j1)) = Sum (H1(D2) | (indx (D2,D1,j1))) by INTEGRA1:def_20;
then H2(D2, indx (D2,D1,j1)) + (Sum (mid ((upper_volume (f,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) = Sum ((H1(D2) | (indx (D2,D1,j1))) ^ (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) by RVSUM_1:75
.= Sum ((mid (H1(D2),1,(indx (D2,D1,j1)))) ^ (mid (H1(D2),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) by A46, FINSEQ_6:116
.= Sum (mid (H1(D2),1,(indx (D2,D1,j)))) by A46, A48, A222, INTEGRA2:4
.= Sum (H1(D2) | (indx (D2,D1,j))) by A129, FINSEQ_6:116 ;
then A226: H2(D2, indx (D2,D1,j1)) + (Sum (mid ((upper_volume (f,D2)),((indx (D2,D1,j1)) + 1),(indx (D2,D1,j))))) = H2(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 H2(D2, indx (D2,D1,j1)) = Sum ((upper_volume (f,D2)) | (indx (D2,D1,j1))) by INTEGRA1:def_20
.= H2(D1,j1) by A218, A42, INTEGRA1:def_20 ;
hence H2(D1,j) - H2(D2, indx (D2,D1,j)) <= (1 * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by A50, A226, A225; ::_thesis: verum
end;
hence S1[1] by A38, A39; ::_thesis: verum
end;
reconsider i = i as non empty Element of NAT by A15, FINSEQ_1:1;
A227: for i being non empty Nat st S1[i] holds
S1[i + 1]
proof
let i be non empty Nat; ::_thesis: ( S1[i] implies S1[i + 1] )
A228: i >= 1 by NAT_1:14;
assume A229: S1[i] ; ::_thesis: S1[i + 1]
S1[i + 1]
proof
A230: i <= i + 1 by NAT_1:11;
assume A231: i + 1 in dom D ; ::_thesis: ex j being Element of NAT st
( j in dom D1 & D . (i + 1) in divset (D1,j) & H2(D1,j) - H2(D2, indx (D2,D1,j)) <= ((i + 1) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) )
then consider j being Element of NAT 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 H1(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: H2(D1,n1) - H2(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 ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( n1 = j or n1 > j ) by A246, XXREAL_0:1;
supposeA247: n1 = j ; ::_thesis: contradiction
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; ::_thesis: verum
end;
suppose n1 > j ; ::_thesis: contradiction
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
percases ( j = 1 or j <> 1 ) ;
suppose j = 1 ; ::_thesis: upper_bound (divset (D1,j)) = D1 . j
hence upper_bound (divset (D1,j)) = D1 . j by A232, INTEGRA1:def_4; ::_thesis: verum
end;
suppose j <> 1 ; ::_thesis: upper_bound (divset (D1,j)) = D1 . j
hence upper_bound (divset (D1,j)) = D1 . j by A232, INTEGRA1:def_4; ::_thesis: verum
end;
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; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
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 H1(D1) by A264, INTEGRA1:def_6;
then j in Seg (len H1(D1)) by A261, FINSEQ_1:1;
then A269: j in dom H1(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 H1(D1)) by INTEGRA1:def_6;
then n1 in dom H1(D1) by FINSEQ_1:def_3;
then H2(D1,n1) = Sum (H1(D1) | n1) by INTEGRA1:def_20
.= Sum (mid (H1(D1),1,n1)) by A259, FINSEQ_6:116 ;
then H2(D1,n1) + (Sum (mid (H1(D1),(n1 + 1),j))) = Sum ((mid (H1(D1),1,n1)) ^ (mid (H1(D1),(n1 + 1),j))) by RVSUM_1:75
.= Sum (mid (H1(D1),1,j)) by A245, A259, A268, INTEGRA2:4
.= Sum (H1(D1) | j) by A261, FINSEQ_6:116 ;
then A272: H2(D1,j) = H2(D1,n1) + (Sum (mid (H1(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 H1(D2)) by INTEGRA1:def_6;
then A273: indx (D2,D1,j) in dom H1(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 (H1(D1),(n1 + 1),j))) - (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
proof
percases ( n1 + 1 = j or n1 + 1 < j ) by A258, XXREAL_0:1;
supposeA279: n1 + 1 = j ; ::_thesis: (Sum (mid (H1(D1),(n1 + 1),j))) - (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
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 ; ::_thesis: contradiction
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 ; ::_thesis: contradiction
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; ::_thesis: verum
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 ; ::_thesis: contradiction
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; ::_thesis: verum
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; ::_thesis: verum
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) ; ::_thesis: (indx (D2,D1,n1)) + 2 = 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 (indx (D2,D1,n1)) + 2 = indx (D2,D1,j) by A308, XXREAL_0:1; ::_thesis: verum
end;
A309: 1 <= (indx (D2,D1,n1)) + 1 by NAT_1:12;
A310: indx (D2,D1,j) <= len H1(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 H1(D2) by INTEGRA1:def_6;
then A313: len (mid (H1(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 (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) >= (lower_bound (rng f)) * (vol (divset (D1,(n1 + 1))))
proof
now__::_thesis:_Sum_(mid_(H1(D2),((indx_(D2,D1,n1))_+_1),(indx_(D2,D1,j))))_>=_(lower_bound_(rng_f))_*_(vol_(divset_(D1,(n1_+_1))))
percases ( (indx (D2,D1,j)) - (indx (D2,D1,n1)) = 1 or (indx (D2,D1,j)) - (indx (D2,D1,n1)) = 2 ) by A314, A307;
supposeA316: (indx (D2,D1,j)) - (indx (D2,D1,n1)) = 1 ; ::_thesis: Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) >= (lower_bound (rng f)) * (vol (divset (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;
1 = ((indx (D2,D1,j)) - ((indx (D2,D1,n1)) + 1)) + 1 by A316;
then (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . 1 = H1(D2) . ((1 + ((indx (D2,D1,n1)) + 1)) - 1) by A309, A310, FINSEQ_6:122
.= H1(D2) . ((indx (D2,D1,n1)) + 1) ;
then A322: mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))) = <*(H1(D2) . ((indx (D2,D1,n1)) + 1))*> by A313, A316, FINSEQ_1:40;
H1(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 H1(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 Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) >= (lower_bound (rng f)) * (vol (divset (D1,(n1 + 1)))) by A322, FINSOP_1:11; ::_thesis: verum
end;
supposeA323: (indx (D2,D1,j)) - (indx (D2,D1,n1)) = 2 ; ::_thesis: Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) >= (lower_bound (rng f)) * (vol (divset (D1,(n1 + 1))))
(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 H1(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 H1(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: H1(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 (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . 2 = H1(D2) . ((2 + ((indx (D2,D1,n1)) + 1)) - 1) by A332, A333, A330, FINSEQ_6:122
.= H1(D2) . (((indx (D2,D1,n1)) + 0) + 2) ;
((indx (D2,D1,j)) - ((indx (D2,D1,n1)) + 1)) + 1 >= 1 by A323;
then (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . 1 = H1(D2) . ((1 + ((indx (D2,D1,n1)) + 1)) - 1) by A323, A332, A333, A330, FINSEQ_6:122
.= H1(D2) . ((indx (D2,D1,n1)) + 1) ;
then mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))) = <*(H1(D2) . ((indx (D2,D1,n1)) + 1)),(H1(D2) . ((indx (D2,D1,n1)) + 2))*> by A313, A323, A336, FINSEQ_1:44;
then A337: Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) = (H1(D2) . ((indx (D2,D1,n1)) + 1)) + (H1(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;
H1(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 H1(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 (H1(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 Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) >= (lower_bound (rng f)) * (vol (divset (D1,(n1 + 1)))) by A279, A340, A326, A328; ::_thesis: verum
end;
end;
end;
hence Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) >= (lower_bound (rng f)) * (vol (divset (D1,(n1 + 1)))) ; ::_thesis: verum
end;
A341: n1 + 1 <= len H1(D1) by A265, INTEGRA1:def_6;
(j -' (n1 + 1)) + 1 = (j - (n1 + 1)) + 1 by A279, XREAL_1:233;
then A342: len (mid (H1(D1),(n1 + 1),j)) = 1 by A244, A279, A341, FINSEQ_6:118;
(n1 + 1) + 1 <= j + 1 by A258, XREAL_1:6;
then 1 <= (j + 1) - (n1 + 1) by XREAL_1:19;
then (mid (H1(D1),(n1 + 1),j)) . 1 = H1(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 (H1(D1),(n1 + 1),j) = <*((upper_bound (rng (f | (divset (D1,(n1 + 1)))))) * (vol (divset (D1,(n1 + 1)))))*> by A342, FINSEQ_1:40;
then A343: Sum (mid (H1(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 (H1(D1),(n1 + 1),j)) <= (upper_bound (rng f)) * (vol (divset (D1,(n1 + 1)))) by A343, A344, XREAL_1:64;
then (Sum (mid (H1(D1),(n1 + 1),j))) - (Sum (mid (H1(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 (Sum (mid (H1(D1),(n1 + 1),j))) - (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) by A347, XXREAL_0:2; ::_thesis: verum
end;
supposeA348: n1 + 1 < j ; ::_thesis: (Sum (mid (H1(D1),(n1 + 1),j))) - (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1)
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
percases ( len MD1 = 1 or len MD1 <> 1 ) ;
supposeA364: len MD1 = 1 ; ::_thesis: ( lower_bound (divset (MD1,(len MD1))) = lower_bound (divset (D1,j)) & upper_bound (divset (MD1,(len MD1))) = upper_bound (divset (D1,j)) )
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 ( lower_bound (divset (MD1,(len MD1))) = lower_bound (divset (D1,j)) & upper_bound (divset (MD1,(len MD1))) = upper_bound (divset (D1,j)) ) by A261, A264, A353, A356, A359, A361, A364, A365, A366, FINSEQ_6:118, INTEGRA1:def_4; ::_thesis: verum
end;
supposeA367: len MD1 <> 1 ; ::_thesis: ( lower_bound (divset (MD1,(len MD1))) = lower_bound (divset (D1,j)) & upper_bound (divset (MD1,(len MD1))) = upper_bound (divset (D1,j)) )
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 ( lower_bound (divset (MD1,(len MD1))) = lower_bound (divset (D1,j)) & upper_bound (divset (MD1,(len MD1))) = upper_bound (divset (D1,j)) ) by A232, A245, A277, A370, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
A372: B c= A
proof
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in B or x1 in A )
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 ; ::_thesis: x1 in A
then reconsider x1 = 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 x1 in A by A378, INTEGRA2:1; ::_thesis: verum
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 being real number such that
A381: for x being set st x in A /\ (dom f) holds
a <= f . x by A1, RFUNCT_1:71;
for x being set st x in B /\ (dom g) holds
a <= g . x
proof
let x be set ; ::_thesis: ( x in B /\ (dom g) implies a <= g . x )
A382: (dom f) /\ B c= (dom f) /\ A by A372, XBOOLE_1:26;
assume x in B /\ (dom g) ; ::_thesis: a <= g . x
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 a <= g . x by A383, FUNCT_1:47; ::_thesis: verum
end;
then A384: g | B is bounded_below by RFUNCT_1:71;
consider a being real number such that
A385: for x being set st x in A /\ (dom f) holds
f . x <= a by A1, RFUNCT_1:70;
for x being set st x in B /\ (dom g) holds
g . x <= a
proof
let x be set ; ::_thesis: ( x in B /\ (dom g) implies g . x <= a )
A386: (dom f) /\ B c= (dom f) /\ A by A372, XBOOLE_1:26;
assume x in B /\ (dom g) ; ::_thesis: g . x <= a
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 g . x <= a by A387, FUNCT_1:47; ::_thesis: verum
end;
then g | B is bounded_above by RFUNCT_1:70;
hence g | B is bounded by A384; ::_thesis: verum
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 H1(D1) by A264, INTEGRA1:def_6;
A392: for k being Nat st 1 <= k & k <= len (upper_volume (g,MD1)) holds
(upper_volume (g,MD1)) . k = (mid (H1(D1),(n1 + 1),j)) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len (upper_volume (g,MD1)) implies (upper_volume (g,MD1)) . k = (mid (H1(D1),(n1 + 1),j)) . k )
assume that
A393: 1 <= k and
A394: k <= len (upper_volume (g,MD1)) ; ::_thesis: (upper_volume (g,MD1)) . k = (mid (H1(D1),(n1 + 1),j)) . k
A395: k in Seg (len (upper_volume (g,MD1))) by A393, A394, FINSEQ_1:1;
then A396: k in Seg (len MD1) by INTEGRA1:def_6;
then A397: k in dom MD1 by FINSEQ_1:def_3;
k in dom MD1 by A396, FINSEQ_1:def_3;
then A398: (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
A399: n1 + 1 = 1 + k2 ;
A400: 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, A399, XXREAL_0:2;
then A401: k + k2 in Seg (len D1) by A400, FINSEQ_1:1;
then A402: k + k2 in dom D1 by FINSEQ_1:def_3;
1 + 1 <= k + k2 by A259, A393, A399, XREAL_1:7;
then A403: 1 < k + k2 by NAT_1:13;
A404: k2 = (n1 + 1) - 1 by A399;
A405: ( lower_bound (divset (D1,(k + k2))) = lower_bound (divset (MD1,k)) & upper_bound (divset (D1,(k + k2))) = upper_bound (divset (MD1,k)) )
proof
percases ( k = 1 or k <> 1 ) ;
supposeA406: k = 1 ; ::_thesis: ( lower_bound (divset (D1,(k + k2))) = lower_bound (divset (MD1,k)) & upper_bound (divset (D1,(k + k2))) = upper_bound (divset (MD1,k)) )
then upper_bound (divset (MD1,k)) = MD1 . k by A397, INTEGRA1:def_4;
then A407: upper_bound (divset (MD1,k)) = D1 . ((k + (n1 + 1)) - 1) by A258, A264, A244, A356, A390, A393, A394, A395, FINSEQ_6:122;
lower_bound (divset (MD1,k)) = D1 . n1 by A353, A397, A406, INTEGRA1:def_4;
hence ( lower_bound (divset (D1,(k + k2))) = lower_bound (divset (MD1,k)) & upper_bound (divset (D1,(k + k2))) = upper_bound (divset (MD1,k)) ) by A404, A403, A402, A406, A407, INTEGRA1:def_4; ::_thesis: verum
end;
supposeA408: k <> 1 ; ::_thesis: ( lower_bound (divset (D1,(k + k2))) = lower_bound (divset (MD1,k)) & upper_bound (divset (D1,(k + k2))) = upper_bound (divset (MD1,k)) )
then upper_bound (divset (MD1,k)) = MD1 . k by A397, INTEGRA1:def_4;
then A409: upper_bound (divset (MD1,k)) = D1 . ((k + (n1 + 1)) - 1) by A258, A264, A244, A356, A390, A393, A394, A395, FINSEQ_6:122;
A410: k - 1 <= (j - (n1 + 1)) + 1 by A390, A394, XREAL_1:146, XXREAL_0:2;
A411: lower_bound (divset (MD1,k)) = MD1 . (k - 1) by A397, A408, INTEGRA1:def_4;
A412: k - 1 in dom MD1 by A397, A408, 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, A412, A410, A411, FINSEQ_6:122;
hence ( lower_bound (divset (D1,(k + k2))) = lower_bound (divset (MD1,k)) & upper_bound (divset (D1,(k + k2))) = upper_bound (divset (MD1,k)) ) by A399, A403, A402, A409, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
divset (MD1,k) = [.(lower_bound (divset (MD1,k))),(upper_bound (divset (MD1,k))).] by INTEGRA1:4;
then A413: divset (D1,(k + k2)) = divset (MD1,k) by A405, INTEGRA1:4;
A414: k + k2 in dom D1 by A401, FINSEQ_1:def_3;
A415: (mid (H1(D1),(n1 + 1),j)) . k = H1(D1) . ((k + (n1 + 1)) - 1) by A258, A244, A390, A391, A393, A394, A395, FINSEQ_6:122
.= (upper_bound (rng (f | (divset (D1,(k + k2)))))) * (vol (divset (D1,(k + k2)))) by A399, A414, INTEGRA1:def_6 ;
k in dom MD1 by A396, FINSEQ_1:def_3;
then divset (D1,(k + k2)) c= B by A413, INTEGRA1:8;
hence (upper_volume (g,MD1)) . k = (mid (H1(D1),(n1 + 1),j)) . k by A398, A415, A413, FUNCT_1:51; ::_thesis: verum
end;
n1 + 1 <= len H1(D1) by A265, INTEGRA1:def_6;
then len (upper_volume (g,MD1)) = len (mid (H1(D1),(n1 + 1),j)) by A258, A261, A244, A389, A390, A391, FINSEQ_6:118;
then A416: Sum (upper_volume (g,MD1)) = Sum (mid (H1(D1),(n1 + 1),j)) by A392, FINSEQ_1:14;
A417: n1 < j - 1 by A348, XREAL_1:20;
A418: 1 <= (indx (D2,D1,n1)) + 1 by A263, NAT_1:13;
A419: len MD1 in dom MD1 by FINSEQ_5:6;
A420: upper_bound (divset (MD1,(len MD1))) = MD1 . (len MD1)
proof
percases ( len MD1 = 1 or len MD1 <> 1 ) ;
suppose len MD1 = 1 ; ::_thesis: upper_bound (divset (MD1,(len MD1))) = MD1 . (len MD1)
hence upper_bound (divset (MD1,(len MD1))) = MD1 . (len MD1) by A419, INTEGRA1:def_4; ::_thesis: verum
end;
suppose len MD1 <> 1 ; ::_thesis: upper_bound (divset (MD1,(len MD1))) = MD1 . (len MD1)
hence upper_bound (divset (MD1,(len MD1))) = MD1 . (len MD1) by A419, INTEGRA1:def_4; ::_thesis: verum
end;
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, A420, INTEGRA1:def_4;
then A421: vol B <> 0 by A232, A241, A245, SEQM_3:def_1;
rng f is bounded_below by A1, INTEGRA1:11;
then A422: 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 A422, XREAL_1:13;
then A423: ((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 A424: (indx (D2,D1,n1)) + 1 <= indx (D2,D1,(n1 + 1)) by NAT_1:13;
then A425: (indx (D2,D1,n1)) + 1 <= len D2 by A274, XXREAL_0:2;
A426: indx (D2,D1,n1) < (indx (D2,D1,n1)) + 1 by NAT_1:13;
A427: indx (D2,D1,(n1 + 1)) = (indx (D2,D1,n1)) + 1
proof
assume indx (D2,D1,(n1 + 1)) <> (indx (D2,D1,n1)) + 1 ; ::_thesis: contradiction
then A428: indx (D2,D1,(n1 + 1)) > (indx (D2,D1,n1)) + 1 by A424, XXREAL_0:1;
A429: (indx (D2,D1,n1)) + 1 in dom D2 by A418, A425, FINSEQ_3:25;
then A430: D2 . ((indx (D2,D1,n1)) + 1) in rng D2 by FUNCT_1:def_3;
now__::_thesis:_contradiction
percases ( D2 . ((indx (D2,D1,n1)) + 1) in rng D1 or D2 . ((indx (D2,D1,n1)) + 1) in rng D ) by A11, A430, XBOOLE_0:def_3;
suppose D2 . ((indx (D2,D1,n1)) + 1) in rng D1 ; ::_thesis: contradiction
then consider n2 being Element of NAT such that
A431: n2 in dom D1 and
A432: D2 . ((indx (D2,D1,n1)) + 1) = D1 . n2 by PARTFUN1:3;
D2 . (indx (D2,D1,n1)) < D2 . ((indx (D2,D1,n1)) + 1) by A262, A426, A429, SEQM_3:def_1;
then n1 < n2 by A241, A260, A431, A432, SEQ_4:137;
then A433: n1 + 1 <= n2 by NAT_1:13;
D1 . n2 < D1 . (n1 + 1) by A270, A267, A428, A429, A432, SEQM_3:def_1;
hence contradiction by A266, A431, A433, SEQ_4:137; ::_thesis: verum
end;
supposeA434: D2 . ((indx (D2,D1,n1)) + 1) in rng D ; ::_thesis: contradiction
A435: D . i <= upper_bound (divset (D1,n1)) by A242, INTEGRA2:1;
A436: upper_bound (divset (D1,n1)) = D1 . n1
proof
percases ( n1 = 1 or n1 <> 1 ) ;
suppose n1 = 1 ; ::_thesis: upper_bound (divset (D1,n1)) = D1 . n1
hence upper_bound (divset (D1,n1)) = D1 . n1 by A241, INTEGRA1:def_4; ::_thesis: verum
end;
suppose n1 <> 1 ; ::_thesis: upper_bound (divset (D1,n1)) = D1 . n1
hence upper_bound (divset (D1,n1)) = D1 . n1 by A241, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
consider n2 being Element of NAT such that
A437: n2 in dom D and
A438: D2 . ((indx (D2,D1,n1)) + 1) = D . n2 by A434, PARTFUN1:3;
D1 . n1 < D . n2 by A262, A260, A426, A429, A438, SEQM_3:def_1;
then D . i < D . n2 by A435, A436, XXREAL_0:2;
then i < n2 by A236, A437, SEQ_4:137;
then A439: i + 1 <= n2 by NAT_1:13;
(n1 + 1) + 1 <= j by A348, NAT_1:13;
then A440: n1 + 1 <= j - 1 by XREAL_1:19;
j - 1 in dom D1 by A232, A245, A277, INTEGRA1:7;
then A441: D1 . (n1 + 1) <= D1 . (j - 1) by A266, A440, SEQ_4:137;
A442: 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 A443: D1 . (n1 + 1) <= D . (i + 1) by A441, A442, XXREAL_0:2;
D . n2 < D1 . (n1 + 1) by A270, A267, A428, A429, A438, SEQM_3:def_1;
then D . n2 < D . (i + 1) by A443, XXREAL_0:2;
hence contradiction by A231, A437, A439, SEQ_4:137; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A444: len MD2 = ((indx (D2,D1,j)) -' (indx (D2,D1,(n1 + 1)))) + 1 by A275, A271, A274, A238, A239, A357, FINSEQ_6:118;
then A445: len MD2 = ((indx (D2,D1,j)) - (indx (D2,D1,(n1 + 1)))) + 1 by A275, XREAL_1:233;
then A446: len (upper_volume (g,MD2)) = ((indx (D2,D1,j)) - ((indx (D2,D1,n1)) + 1)) + 1 by A427, INTEGRA1:def_6;
for x1 being set st x1 in (rng MD1) \/ {(D . (i + 1))} holds
x1 in rng MD2
proof
let x1 be set ; ::_thesis: ( x1 in (rng MD1) \/ {(D . (i + 1))} implies x1 in rng MD2 )
assume A447: x1 in (rng MD1) \/ {(D . (i + 1))} ; ::_thesis: x1 in rng MD2
then reconsider x1 = x1 as Real ;
now__::_thesis:_x1_in_rng_MD2
percases ( x1 in rng MD1 or x1 in {(D . (i + 1))} ) by A447, XBOOLE_0:def_3;
supposeA448: x1 in rng MD1 ; ::_thesis: x1 in rng MD2
rng MD1 <> {} ;
then 1 in dom MD1 by FINSEQ_3:32;
then A449: 1 <= len MD1 by FINSEQ_3:25;
A450: 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 A451: MD1 . (len MD1) = D1 . j by A258, A264, A244, A356, A449, A450, FINSEQ_6:122;
rng MD1 c= rng D1 by A356, FINSEQ_6:119;
then A452: x1 in rng D1 by A448;
rng D1 c= rng D2 by A10, INTEGRA1:def_18;
then consider k being Element of NAT such that
A453: k in dom D2 and
A454: D2 . k = x1 by A452, PARTFUN1:3;
x1 <= MD1 . (len MD1) by A448, Th16;
then k <= indx (D2,D1,j) by A237, A234, A451, A453, A454, 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;
A456: MD1 . 1 <= x1 by A448, Th16;
MD1 . 1 = D1 . (n1 + 1) by A261, A264, A244, A265, A356, FINSEQ_6:118;
then A457: indx (D2,D1,(n1 + 1)) <= k by A270, A267, A456, A453, A454, SEQM_3:def_1;
then consider n being Nat such that
A458: k + 1 = (indx (D2,D1,(n1 + 1))) + n by NAT_1:10, NAT_1:12;
A459: (n + (indx (D2,D1,(n1 + 1)))) - 1 = k by A458;
A460: n in NAT by ORDINAL1:def_12;
(indx (D2,D1,(n1 + 1))) + 1 <= k + 1 by A457, XREAL_1:6;
then A461: 1 <= (k + 1) - (indx (D2,D1,(n1 + 1))) by XREAL_1:19;
then n in dom MD2 by A445, A455, A458, FINSEQ_3:25;
then MD2 . n in rng MD2 by FUNCT_1:def_3;
hence x1 in rng MD2 by A275, A271, A239, A357, A454, A461, A455, A460, A459, FINSEQ_6:122; ::_thesis: verum
end;
suppose x1 in {(D . (i + 1))} ; ::_thesis: x1 in rng MD2
then A462: x1 = D . (i + 1) by TARSKI:def_1;
reconsider j1 = j - 1 as Element of NAT by A232, A245, A277, INTEGRA1:7;
A463: 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 being Element of NAT such that
A464: k in dom D2 and
A465: x1 = D2 . k by A462, A463, PARTFUN1:3;
D . (i + 1) <= upper_bound (divset (D1,j)) by A233, INTEGRA2:1;
then x1 <= D1 . j by A232, A245, A277, A462, INTEGRA1:def_4;
then A466: D2 . k <= D2 . (indx (D2,D1,j)) by A10, A232, A465, INTEGRA1:def_19;
n1 < j1 by A348, XREAL_1:20;
then A467: n1 + 1 <= j1 by NAT_1:13;
j - 1 in dom D1 by A232, A245, A277, INTEGRA1:7;
then A468: D1 . (n1 + 1) <= D1 . (j - 1) by A266, A467, SEQ_4:137;
lower_bound (divset (D1,j)) <= D . (i + 1) by A233, INTEGRA2:1;
then D1 . (j - 1) <= x1 by A232, A245, A277, A462, INTEGRA1:def_4;
then D2 . (indx (D2,D1,(n1 + 1))) <= D2 . k by A267, A465, A468, XXREAL_0:2;
hence x1 in rng MD2 by A270, A237, A357, A464, A465, A466, Th17; ::_thesis: verum
end;
end;
end;
hence x1 in rng MD2 ; ::_thesis: verum
end;
then A469: (rng MD1) \/ {(D . (i + 1))} c= rng MD2 by TARSKI:def_3;
rng MD2 <> {} ;
then 1 in dom MD2 by FINSEQ_3:32;
then A470: 1 <= len MD2 by FINSEQ_3:25;
A471: ((len MD2) - 1) + (indx (D2,D1,(n1 + 1))) = indx (D2,D1,j) by A445;
for x1 being set st x1 in rng MD2 holds
x1 in (rng MD1) \/ {(D . (i + 1))}
proof
let x1 be set ; ::_thesis: ( x1 in rng MD2 implies x1 in (rng MD1) \/ {(D . (i + 1))} )
assume A472: x1 in rng MD2 ; ::_thesis: x1 in (rng MD1) \/ {(D . (i + 1))}
then reconsider x1 = x1 as Real ;
A473: MD2 . 1 <= x1 by A472, Th16;
A474: MD2 . (len MD2) = D2 . (indx (D2,D1,j)) by A275, A271, A239, A357, A470, A444, A471, FINSEQ_6:122;
A475: rng MD2 c= rng D2 by A357, FINSEQ_6:119;
A476: MD2 . 1 = D2 . (indx (D2,D1,(n1 + 1))) by A271, A274, A238, A239, A357, FINSEQ_6:118;
A477: x1 <= MD2 . (len MD2) by A472, Th16;
then A478: x1 <= D1 . j by A234, A275, A271, A239, A357, A470, A444, A471, FINSEQ_6:122;
now__::_thesis:_x1_in_(rng_MD1)_\/_{(D_._(i_+_1))}
percases ( x1 in rng D1 or x1 in rng D ) by A11, A472, A475, XBOOLE_0:def_3;
suppose x1 in rng D1 ; ::_thesis: x1 in (rng MD1) \/ {(D . (i + 1))}
then consider k being Element of NAT such that
A479: k in dom D1 and
A480: D1 . k = x1 by PARTFUN1:3;
A481: n1 + 1 <= k by A266, A267, A473, A476, A479, A480, SEQM_3:def_1;
then A482: 1 <= k - n1 by XREAL_1:19;
n1 <= n1 + 1 by NAT_1:11;
then consider n being Nat such that
A483: k = n1 + n by A481, NAT_1:10, XXREAL_0:2;
A484: k <= j by A232, A234, A477, A474, A479, A480, SEQM_3:def_1;
then A485: k - n1 <= j - n1 by XREAL_1:9;
A486: 1 <= k - n1 by A481, XREAL_1:19;
A487: (j - (n1 + 1)) + 1 = j - n1 ;
k - n1 <= len MD1 by A358, A350, A484, XREAL_1:9;
then n in dom MD1 by A486, A483, FINSEQ_3:25;
then A488: MD1 . n in rng MD1 by FUNCT_1:def_3;
n in NAT by ORDINAL1:def_12;
then MD1 . n = D1 . (((k - n1) - 1) + (n1 + 1)) by A258, A264, A244, A356, A482, A485, A487, A483, FINSEQ_6:122
.= D1 . k ;
hence x1 in (rng MD1) \/ {(D . (i + 1))} by A480, A488, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose x1 in rng D ; ::_thesis: x1 in (rng MD1) \/ {(D . (i + 1))}
then consider n being Element of NAT such that
A489: n in dom D and
A490: D . n = x1 by PARTFUN1:3;
A491: not i + 1 < n
proof
A492: upper_bound (divset (D1,j)) = D1 . j
proof
percases ( j = 1 or j <> 1 ) ;
suppose j = 1 ; ::_thesis: upper_bound (divset (D1,j)) = D1 . j
hence upper_bound (divset (D1,j)) = D1 . j by A232, INTEGRA1:def_4; ::_thesis: verum
end;
suppose j <> 1 ; ::_thesis: upper_bound (divset (D1,j)) = D1 . j
hence upper_bound (divset (D1,j)) = D1 . j by A232, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
consider y1 being Real such that
A493: y1 = D . (i + 1) ;
A494: D . n in rng D by A489, FUNCT_1:def_3;
assume i + 1 < n ; ::_thesis: contradiction
then A495: D . (i + 1) < D . n by A231, A489, 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 A495, XXREAL_0:2;
then D . n in divset (D1,j) by A478, A490, A492, INTEGRA2:1;
then A496: x1 in (rng D) /\ (divset (D1,j)) by A490, A494, 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, A493, XBOOLE_0:def_4;
hence contradiction by A8, A232, A490, A495, A496, A493, Th5; ::_thesis: verum
end;
A497: upper_bound (divset (D1,n1)) = D1 . n1
proof
percases ( n1 = 1 or n1 <> 1 ) ;
suppose n1 = 1 ; ::_thesis: upper_bound (divset (D1,n1)) = D1 . n1
hence upper_bound (divset (D1,n1)) = D1 . n1 by A241, INTEGRA1:def_4; ::_thesis: verum
end;
suppose n1 <> 1 ; ::_thesis: upper_bound (divset (D1,n1)) = D1 . n1
hence upper_bound (divset (D1,n1)) = D1 . n1 by A241, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
D . i <= upper_bound (divset (D1,n1)) by A242, INTEGRA2:1;
then D . i < D1 . (n1 + 1) by A352, A497, XXREAL_0:2;
then D . i < D . n by A267, A473, A476, A490, XXREAL_0:2;
then i < n by A236, A489, 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 A490, A491, TARSKI:def_1;
hence x1 in (rng MD1) \/ {(D . (i + 1))} by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
hence x1 in (rng MD1) \/ {(D . (i + 1))} ; ::_thesis: verum
end;
then rng MD2 c= (rng MD1) \/ {(D . (i + 1))} by TARSKI:def_3;
then A498: rng MD2 = (rng MD1) \/ {(D . (i + 1))} by A469, XBOOLE_0:def_10;
delta MD1 in rng (upper_volume ((chi (B,B)),MD1)) by XXREAL_2:def_8;
then consider k being Element of NAT such that
A499: k in dom (upper_volume ((chi (B,B)),MD1)) and
A500: (upper_volume ((chi (B,B)),MD1)) . k = delta MD1 by PARTFUN1:3;
A501: k in Seg (len (upper_volume ((chi (B,B)),MD1))) by A499, FINSEQ_1:def_3;
then A502: k in Seg (len MD1) by INTEGRA1:def_6;
then A503: k in dom MD1 by FINSEQ_1:def_3;
A504: k <= len MD1 by A502, FINSEQ_1:1;
then k + n1 <= j by A358, A350, XREAL_1:19;
then A505: k + n1 <= len D1 by A264, XXREAL_0:2;
A506: 1 <= k by A501, FINSEQ_1:1;
A507: n1 + 1 > 1 by A277, NAT_1:13;
then n1 > 1 - 1 by XREAL_1:19;
then A508: k < k + n1 by XREAL_1:29;
then 1 < k + n1 by A506, XXREAL_0:2;
then A509: k + n1 in dom D1 by A505, 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
percases ( k = 1 or k <> 1 ) ;
supposeA510: k = 1 ; ::_thesis: ( lower_bound (divset (MD1,k)) = lower_bound (divset (D1,(k + n1))) & upper_bound (divset (MD1,k)) = upper_bound (divset (D1,(k + n1))) )
then upper_bound (divset (MD1,k)) = MD1 . k by A503, INTEGRA1:def_4;
then A511: upper_bound (divset (MD1,k)) = D1 . ((k + (n1 + 1)) - 1) by A258, A264, A244, A356, A358, A506, A504, FINSEQ_6:122;
lower_bound (divset (D1,(k + n1))) = D1 . ((k + n1) - 1) by A506, A508, A509, INTEGRA1:def_4;
hence ( lower_bound (divset (MD1,k)) = lower_bound (divset (D1,(k + n1))) & upper_bound (divset (MD1,k)) = upper_bound (divset (D1,(k + n1))) ) by A353, A507, A503, A509, A510, A511, INTEGRA1:def_4; ::_thesis: verum
end;
supposeA512: k <> 1 ; ::_thesis: ( lower_bound (divset (MD1,k)) = lower_bound (divset (D1,(k + n1))) & upper_bound (divset (MD1,k)) = upper_bound (divset (D1,(k + n1))) )
then upper_bound (divset (MD1,k)) = MD1 . k by A503, INTEGRA1:def_4;
then A513: upper_bound (divset (MD1,k)) = D1 . ((k + (n1 + 1)) - 1) by A258, A264, A244, A356, A358, A506, A504, FINSEQ_6:122;
A514: lower_bound (divset (MD1,k)) = MD1 . (k - 1) by A503, A512, INTEGRA1:def_4;
A515: k - 1 in dom MD1 by A503, A512, INTEGRA1:7;
then A516: k - 1 <= len MD1 by FINSEQ_3:25;
1 <= k - 1 by A515, FINSEQ_3:25;
then lower_bound (divset (MD1,k)) = D1 . (((k - 1) + (n1 + 1)) - 1) by A258, A264, A244, A356, A358, A515, A516, A514, FINSEQ_6:122;
hence ( lower_bound (divset (MD1,k)) = lower_bound (divset (D1,(k + n1))) & upper_bound (divset (MD1,k)) = upper_bound (divset (D1,(k + n1))) ) by A506, A508, A509, A513, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
then divset (MD1,k) = [.(lower_bound (divset (D1,(k + n1)))),(upper_bound (divset (D1,(k + n1)))).] by INTEGRA1:4;
then A517: divset (MD1,k) = divset (D1,(k + n1)) by INTEGRA1:4;
k + n1 in Seg (len D1) by A509, FINSEQ_1:def_3;
then k + n1 in Seg (len (upper_volume ((chi (A,A)),D1))) by INTEGRA1:def_6;
then A518: k + n1 in dom (upper_volume ((chi (A,A)),D1)) by FINSEQ_1:def_3;
k in dom MD1 by A502, FINSEQ_1:def_3;
then delta MD1 = vol (divset (MD1,k)) by A500, INTEGRA1:20;
then delta MD1 = (upper_volume ((chi (A,A)),D1)) . (k + n1) by A509, A517, INTEGRA1:20;
then delta MD1 in rng (upper_volume ((chi (A,A)),D1)) by A518, FUNCT_1:def_3;
then delta MD1 <= max (rng (upper_volume ((chi (A,A)),D1))) by XXREAL_2:def_8;
then A519: delta MD1 <= delta D1 ;
A520: 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 A521: D . (i + 1) in divset (MD1,(len MD1)) by A363, A520, INTEGRA2:1;
j - 1 in dom D1 by A232, A245, A277, INTEGRA1:7;
then D1 . n1 < D1 . (j - 1) by A241, A417, 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, A498, A521, A421, Th14;
then A522: (Sum (upper_volume (g,MD1))) - (Sum (upper_volume (g,MD2))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta MD1) by A423, XXREAL_0:2;
A523: indx (D2,D1,j) <= len H1(D2) by A239, INTEGRA1:def_6;
A524: (indx (D2,D1,n1)) + 1 <= indx (D2,D1,j) by A275, A424, XXREAL_0:2;
A525: for k being Nat st 1 <= k & k <= len (upper_volume (g,MD2)) holds
(upper_volume (g,MD2)) . k = (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len (upper_volume (g,MD2)) implies (upper_volume (g,MD2)) . k = (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . k )
assume that
A526: 1 <= k and
A527: k <= len (upper_volume (g,MD2)) ; ::_thesis: (upper_volume (g,MD2)) . k = (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . k
A528: k in Seg (len (upper_volume (g,MD2))) by A526, A527, FINSEQ_1:1;
then A529: (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . k = H1(D2) . ((k + ((indx (D2,D1,n1)) + 1)) - 1) by A418, A446, A523, A524, A526, A527, FINSEQ_6:122;
A530: k in Seg (len MD2) by A528, INTEGRA1:def_6;
then k in dom MD2 by FINSEQ_1:def_3;
then A531: (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 A526, XREAL_1:7;
then A532: 1 <= (k + ((indx (D2,D1,n1)) + 1)) - 1 by XREAL_1:19;
consider k2 being Element of NAT such that
A533: (indx (D2,D1,n1)) + 1 = 1 + k2 ;
k <= (indx (D2,D1,j)) - (((indx (D2,D1,n1)) + 1) - 1) by A445, A427, A527, 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 H1(D2) by A523, XXREAL_0:2;
then k + k2 in Seg (len H1(D2)) by A532, A533, FINSEQ_1:1;
then A534: k + k2 in Seg (len D2) by INTEGRA1:def_6;
then k + k2 in dom D2 by FINSEQ_1:def_3;
then A535: (mid (H1(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 A529, A533, INTEGRA1:def_6;
A536: ( 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, A526, A533, XREAL_1:7;
then A537: k + k2 > 1 by NAT_1:13;
A538: k in dom MD2 by A530, FINSEQ_1:def_3;
A539: k + k2 in dom D2 by A534, FINSEQ_1:def_3;
percases ( k = 1 or k <> 1 ) ;
supposeA540: k = 1 ; ::_thesis: ( lower_bound (divset (MD2,k)) = lower_bound (divset (D2,(k + k2))) & upper_bound (divset (MD2,k)) = upper_bound (divset (D2,(k + k2))) )
then A541: upper_bound (divset (D2,(k + k2))) = D2 . (1 + k2) by A537, A539, INTEGRA1:def_4;
A542: lower_bound (divset (MD2,k)) = lower_bound B by A538, A540, INTEGRA1:def_4;
upper_bound (divset (MD2,k)) = MD2 . k by A538, A540, INTEGRA1:def_4;
then A543: upper_bound (divset (MD2,k)) = D2 . ((1 + (indx (D2,D1,(n1 + 1)))) - 1) by A275, A239, A357, A418, A427, A446, A527, A540, FINSEQ_6:122
.= D1 . (n1 + 1) by A10, A266, INTEGRA1:def_19 ;
lower_bound (divset (D2,(k + k2))) = D2 . ((1 + k2) - 1) by A537, A539, A540, INTEGRA1:def_4;
hence ( lower_bound (divset (MD2,k)) = lower_bound (divset (D2,(k + k2))) & upper_bound (divset (MD2,k)) = upper_bound (divset (D2,(k + k2))) ) by A10, A241, A266, A353, A427, A533, A542, A543, A541, INTEGRA1:def_19; ::_thesis: verum
end;
supposeA544: k <> 1 ; ::_thesis: ( lower_bound (divset (MD2,k)) = lower_bound (divset (D2,(k + k2))) & upper_bound (divset (MD2,k)) = upper_bound (divset (D2,(k + k2))) )
then upper_bound (divset (MD2,k)) = MD2 . k by A538, INTEGRA1:def_4;
then A545: upper_bound (divset (MD2,k)) = D2 . ((k + ((indx (D2,D1,n1)) + 1)) - 1) by A275, A239, A357, A418, A427, A446, A526, A527, A528, FINSEQ_6:122;
A546: k - 1 <= ((indx (D2,D1,j)) - ((indx (D2,D1,n1)) + 1)) + 1 by A446, A527, XREAL_1:146, XXREAL_0:2;
A547: lower_bound (divset (MD2,k)) = MD2 . (k - 1) by A538, A544, INTEGRA1:def_4;
A548: k - 1 in dom MD2 by A538, A544, 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, A418, A427, A548, A546, A547, FINSEQ_6:122;
hence ( lower_bound (divset (MD2,k)) = lower_bound (divset (D2,(k + k2))) & upper_bound (divset (MD2,k)) = upper_bound (divset (D2,(k + k2))) ) by A533, A537, A539, A545, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
divset (MD2,k) = [.(lower_bound (divset (MD2,k))),(upper_bound (divset (MD2,k))).] by INTEGRA1:4;
then A549: divset (MD2,k) = divset (D2,(k + k2)) by A536, INTEGRA1:4;
k in dom MD2 by A530, FINSEQ_1:def_3;
then divset (D2,(k + k2)) c= B by A549, INTEGRA1:8;
hence (upper_volume (g,MD2)) . k = (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) . k by A531, A535, A549, FUNCT_1:51; ::_thesis: verum
end;
(indx (D2,D1,n1)) + 1 <= len H1(D2) by A425, INTEGRA1:def_6;
then len (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) = ((indx (D2,D1,j)) -' ((indx (D2,D1,n1)) + 1)) + 1 by A238, A418, A523, A524, FINSEQ_6:118;
then len (upper_volume (g,MD2)) = len (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) by A275, A424, A446, XREAL_1:233, XXREAL_0:2;
then A550: Sum (upper_volume (g,MD2)) = Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))) by A525, 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 A519, XREAL_1:64;
hence (Sum (mid (H1(D1),(n1 + 1),j))) - (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) by A522, A550, A416, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
then A551: (H2(D1,n1) - H2(D2, indx (D2,D1,n1))) + ((Sum (mid (H1(D1),(n1 + 1),j))) - (Sum (mid (H1(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 A552: 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 H1(D2)) by INTEGRA1:def_6;
then indx (D2,D1,n1) in dom H1(D2) by FINSEQ_1:def_3;
then H2(D2, indx (D2,D1,n1)) = Sum (H1(D2) | (indx (D2,D1,n1))) by INTEGRA1:def_20
.= Sum (mid (H1(D2),1,(indx (D2,D1,n1)))) by A263, FINSEQ_6:116 ;
then H2(D2, indx (D2,D1,n1)) + (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) = Sum ((mid (H1(D2),1,(indx (D2,D1,n1)))) ^ (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) by RVSUM_1:75
.= Sum (mid (H1(D2),1,(indx (D2,D1,j)))) by A263, A552, A240, INTEGRA2:4
.= Sum (H1(D2) | (indx (D2,D1,j))) by A238, FINSEQ_6:116 ;
then H2(D2, indx (D2,D1,j)) = H2(D2, indx (D2,D1,n1)) + (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j))))) by A273, INTEGRA1:def_20;
then (H2(D1,n1) - H2(D2, indx (D2,D1,n1))) + ((Sum (mid (H1(D1),(n1 + 1),j))) - (Sum (mid (H1(D2),((indx (D2,D1,n1)) + 1),(indx (D2,D1,j)))))) = H2(D1,j) - H2(D2, indx (D2,D1,j)) by A272;
hence ex j being Element of NAT st
( j in dom D1 & D . (i + 1) in divset (D1,j) & H2(D1,j) - H2(D2, indx (D2,D1,j)) <= ((i + 1) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) ) by A232, A233, A551; ::_thesis: verum
end;
hence S1[i + 1] ; ::_thesis: verum
end;
for k being non empty Nat holds S1[k] from NAT_1:sch_10(A36, A227);
then S1[i] ;
hence ex j being Element of NAT st
( j in dom D1 & D . i in divset (D1,j) & H2(D1,j) - H2(D2, indx (D2,D1,j)) <= (i * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) ) by A14; ::_thesis: verum
end;
A553: 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 A554: 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 being Element of NAT such that
A555: j in dom D1 and
A556: D . (len D) in divset (D1,j) and
A557: H2(D1,j) - H2(D2, indx (D2,D1,j)) <= ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by A13;
A558: j = len D1
proof
j in Seg (len D1) by A555, FINSEQ_1:def_3;
then A559: j <= len D1 by FINSEQ_1:1;
assume j <> len D1 ; ::_thesis: contradiction
then j < len D1 by A559, XXREAL_0:1;
then D1 . j < D1 . (len D1) by A555, A553, SEQM_3:def_1;
then A560: D1 . j < upper_bound A by INTEGRA1:def_2;
A561: upper_bound (divset (D1,j)) < upper_bound A
proof
percases ( j = 1 or j <> 1 ) ;
suppose j = 1 ; ::_thesis: upper_bound (divset (D1,j)) < upper_bound A
hence upper_bound (divset (D1,j)) < upper_bound A by A555, A560, INTEGRA1:def_4; ::_thesis: verum
end;
suppose j <> 1 ; ::_thesis: upper_bound (divset (D1,j)) < upper_bound A
hence upper_bound (divset (D1,j)) < upper_bound A by A555, A560, INTEGRA1:def_4; ::_thesis: verum
end;
end;
end;
D . (len D) <= upper_bound (divset (D1,j)) by A556, INTEGRA2:1;
hence contradiction by A561, INTEGRA1:def_2; ::_thesis: verum
end;
indx (D2,D1,(len D1)) in dom D2 by A10, A553, INTEGRA1:def_19;
then indx (D2,D1,(len D1)) = len D2 by A12, A554, SEQ_4:138;
then (upper_sum (f,D1)) - H2(D2, len D2) <= ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by A557, A558, INTEGRA1:42;
hence (upper_sum (f,D1)) - (upper_sum (f,D2)) <= ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by INTEGRA1:42; ::_thesis: verum
end;
hence ex D2 being Division of A 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 A9, A10, A11; ::_thesis: verum
end;
hence ex D2 being Division of A 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) ) ; ::_thesis: verum
end;
assume A562: ( delta T is 0 -convergent & delta T is non-zero ) ; ::_thesis: ( not vol A <> 0 or ( upper_sum (f,T) is convergent & lim (upper_sum (f,T)) = upper_integral f ) )
then A563: delta T is convergent by FDIFF_1:def_1;
A564: lim (delta T) = 0 by A562, FDIFF_1:def_1;
assume A565: vol A <> 0 ; ::_thesis: ( upper_sum (f,T) is convergent & lim (upper_sum (f,T)) = upper_integral f )
A566: delta T is non-zero by A562;
A567: for e being Real st e > 0 holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
( 0 < (delta T) . m & (delta T) . m < e )
proof
let e be Real; ::_thesis: ( e > 0 implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
( 0 < (delta T) . m & (delta T) . m < e ) )
assume e > 0 ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
( 0 < (delta T) . m & (delta T) . m < e )
then consider n being Element of NAT such that
A568: for m being Element of NAT st n <= m holds
abs (((delta T) . m) - 0) < e by A563, A564, SEQ_2:def_7;
take n ; ::_thesis: for m being Element of NAT st n <= m holds
( 0 < (delta T) . m & (delta T) . m < e )
let m be Element of NAT ; ::_thesis: ( n <= m implies ( 0 < (delta T) . m & (delta T) . m < e ) )
assume n <= m ; ::_thesis: ( 0 < (delta T) . m & (delta T) . m < e )
then abs (((delta T) . m) - 0) < e by A568;
then A569: ((delta T) . m) + (abs (((delta T) . m) - 0)) < e + (abs (((delta T) . m) - 0)) by ABSVALUE:4, XREAL_1:8;
reconsider D = T . m as Division of A ;
A570: (delta T) . m = delta (T . m) by Def2;
delta (T . m) in rng (upper_volume ((chi (A,A)),(T . m))) by XXREAL_2:def_8;
then consider i being Element of NAT such that
A571: i in dom (upper_volume ((chi (A,A)),(T . m))) and
A572: delta (T . m) = (upper_volume ((chi (A,A)),(T . m))) . i by PARTFUN1:3;
i in Seg (len (upper_volume ((chi (A,A)),(T . m)))) by A571, 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 A573: delta (T . m) = vol (divset ((T . m),i)) by A572, INTEGRA1:20;
(delta T) . m <> 0 by A566, SEQ_1:5;
hence ( 0 < (delta T) . m & (delta T) . m < e ) by A569, A570, A573, INTEGRA1:9, XREAL_1:6; ::_thesis: verum
end;
A574: for e being real number st e > 0 holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((upper_sum (f,T)) . m) - (upper_integral f)) < e
proof
let e be real number ; ::_thesis: ( e > 0 implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((upper_sum (f,T)) . m) - (upper_integral f)) < e )
assume A575: e > 0 ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((upper_sum (f,T)) . m) - (upper_integral f)) < e
then A576: e / 2 > 0 by XREAL_1:139;
reconsider e = e as Real by XREAL_0:def_1;
A577: 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 being real number such that
A578: y in rng (upper_sum_set f) and
A579: (upper_integral f) + (e / 2) > y by A576, A577, 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
A580: D3 in dom (upper_sum_set f) and
A581: y = (upper_sum_set f) . D3 by A578, PARTFUN1:3;
reconsider D3 = D3 as Division of A by INTEGRA1:def_3;
A582: 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 A583: 1 in dom D3 by FINSEQ_1:def_3;
percases ( D3 . 1 <> lower_bound A or D3 . 1 = lower_bound A ) ;
supposeA584: D3 . 1 <> lower_bound A ; ::_thesis: 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 )
D3 . 1 in A by A583, INTEGRA1:6;
then lower_bound A <= D3 . 1 by INTEGRA2:1;
then D3 . 1 > lower_bound A by A584, XXREAL_0:1;
hence 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 ) by A580, A581; ::_thesis: verum
end;
supposeA585: D3 . 1 = lower_bound A ; ::_thesis: 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 )
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
A586: (upper_volume (f,D3)) . 1 = (upper_bound (rng (f | (divset (D3,1))))) * (vol (divset (D3,1))) by A583, INTEGRA1:def_6;
vol A >= 0 by INTEGRA1:9;
then A587: (upper_bound A) - (lower_bound A) > 0 by A565, INTEGRA1:def_5;
A588: y = upper_sum (f,D3) by A581, 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 ;
A589: D3 . (len D3) = upper_bound A by INTEGRA1:def_2;
len D3 in dom D3 by A582, FINSEQ_1:def_3;
then A590: len D3 > 1 by A583, A585, A589, A587, SEQ_4:137, XREAL_1:47;
then reconsider D = D3 /^ 1 as increasing FinSequence of REAL by INTEGRA1:34;
A591: len D = (len D3) - 1 by A590, RFINSEQ:def_1;
upper_bound A > lower_bound A by A587, XREAL_1:47;
then len D <> 0 by A585, A591, INTEGRA1:def_2;
then reconsider D = D as non empty increasing FinSequence of REAL ;
A592: len D in dom D by FINSEQ_5:6;
(len D) + 1 = len D3 by A591;
then A593: D . (len D) = upper_bound A by A589, A590, A592, RFINSEQ:def_1;
A594: len D in Seg (len D) by FINSEQ_1:3;
1 + 1 <= len D3 by A590, NAT_1:13;
then 2 in dom D3 by FINSEQ_3:25;
then A595: D3 . 1 < D3 . 2 by A583, SEQM_3:def_1;
A596: rng D3 c= A by INTEGRA1:def_2;
rng D c= rng D3 by FINSEQ_5:33;
then rng D c= A by A596, XBOOLE_1:1;
then reconsider D = D as Division of A by A593, INTEGRA1:def_2;
A597: 1 in Seg 1 by FINSEQ_1:1;
A598: len D3 >= 1 + 1 by A590, NAT_1:13;
then A599: 2 <= len (upper_volume (f,D3)) by INTEGRA1:def_6;
1 <= len (upper_volume (f,D3)) by A590, INTEGRA1:def_6;
then A600: len (mid ((upper_volume (f,D3)),2,(len (upper_volume (f,D3))))) = ((len (upper_volume (f,D3))) -' 2) + 1 by A599, FINSEQ_6:118
.= ((len D3) -' 2) + 1 by INTEGRA1:def_6
.= ((len D3) - 2) + 1 by A598, XREAL_1:233
.= (len D3) - 1 ;
A601: for i being 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; ::_thesis: ( 1 <= i & i <= len (mid ((upper_volume (f,D3)),2,(len (upper_volume (f,D3))))) implies (mid ((upper_volume (f,D3)),2,(len (upper_volume (f,D3))))) . i = (upper_volume (f,D)) . i )
assume that
A602: 1 <= i and
A603: i <= len (mid ((upper_volume (f,D3)),2,(len (upper_volume (f,D3))))) ; ::_thesis: (mid ((upper_volume (f,D3)),2,(len (upper_volume (f,D3))))) . i = (upper_volume (f,D)) . i
A604: 1 <= i + 1 by NAT_1:12;
i + 1 <= len D3 by A600, A603, XREAL_1:19;
then A605: i + 1 in Seg (len D3) by A604, FINSEQ_1:1;
then A606: i + 1 in dom D3 by FINSEQ_1:def_3;
A607: divset (D3,(i + 1)) = divset (D,i)
proof
A608: i + 1 in dom D3 by A605, FINSEQ_1:def_3;
A609: 1 <> i + 1 by A602, NAT_1:13;
then A610: upper_bound (divset (D3,(i + 1))) = D3 . (i + 1) by A608, INTEGRA1:def_4;
A611: i in dom D by A591, A600, A602, A603, FINSEQ_3:25;
then A612: D . i = D3 . (i + 1) by A590, RFINSEQ:def_1;
A613: lower_bound (divset (D3,(i + 1))) = D3 . ((i + 1) - 1) by A609, A608, INTEGRA1:def_4;
percases ( i = 1 or i <> 1 ) ;
supposeA614: i = 1 ; ::_thesis: divset (D3,(i + 1)) = divset (D,i)
then A615: upper_bound (divset (D,i)) = D . i by A611, INTEGRA1:def_4;
A616: lower_bound (divset (D,i)) = lower_bound A by A611, A614, INTEGRA1:def_4;
divset (D3,(i + 1)) = [.(lower_bound A),(D . i).] by A585, A610, A613, A612, A614, INTEGRA1:4;
hence divset (D3,(i + 1)) = divset (D,i) by A616, A615, INTEGRA1:4; ::_thesis: verum
end;
supposeA617: i <> 1 ; ::_thesis: divset (D3,(i + 1)) = divset (D,i)
then i - 1 in dom D by A611, INTEGRA1:7;
then A618: D . (i - 1) = D3 . ((i - 1) + 1) by A590, RFINSEQ:def_1
.= D3 . i ;
A619: upper_bound (divset (D,i)) = D . i by A611, A617, INTEGRA1:def_4;
lower_bound (divset (D,i)) = D . (i - 1) by A611, A617, INTEGRA1:def_4;
then divset (D3,(i + 1)) = [.(lower_bound (divset (D,i))),(upper_bound (divset (D,i))).] by A610, A613, A612, A619, A618, INTEGRA1:4;
hence divset (D3,(i + 1)) = divset (D,i) by INTEGRA1:4; ::_thesis: verum
end;
end;
end;
i <= (len (upper_volume (f,D3))) - 1 by A600, A603, INTEGRA1:def_6;
then A620: i <= ((len (upper_volume (f,D3))) - 2) + 1 ;
i in NAT by ORDINAL1:def_12;
then (mid ((upper_volume (f,D3)),2,(len (upper_volume (f,D3))))) . i = (upper_volume (f,D3)) . ((i + 2) - 1) by A599, A602, A620, FINSEQ_6:122
.= (upper_volume (f,D3)) . (i + 1) ;
then A621: (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 A606, INTEGRA1:def_6;
i in Seg (len D) by A591, A600, A602, A603, FINSEQ_1:1;
then i in dom D by FINSEQ_1:def_3;
hence (mid ((upper_volume (f,D3)),2,(len (upper_volume (f,D3))))) . i = (upper_volume (f,D)) . i by A621, A607, INTEGRA1:def_6; ::_thesis: verum
end;
A622: 1 <= len (upper_volume (f,D3)) by A590, INTEGRA1:def_6;
then A623: len ((upper_volume (f,D3)) | 1) = 1 by FINSEQ_1:59;
1 in dom (upper_volume (f,D3)) by A622, FINSEQ_3:25;
then ((upper_volume (f,D3)) | 1) . 1 = (upper_volume (f,D3)) . 1 by A597, RFINSEQ:6;
then A624: (upper_volume (f,D3)) | 1 = <*((upper_volume (f,D3)) . 1)*> by A623, FINSEQ_1:40;
A625: 2 -' 1 = 2 - 1 by XREAL_1:233
.= 1 ;
1 <= len D by A594, FINSEQ_1:1;
then 1 in dom D by FINSEQ_3:25;
then A626: D . 1 = D3 . (1 + 1) by A590, RFINSEQ:def_1
.= D3 . 2 ;
D in divs A by INTEGRA1:def_3;
then A627: D in dom (upper_sum_set f) by FUNCT_2:def_1;
len (upper_volume (f,D3)) >= 2 by A598, INTEGRA1:def_6;
then A628: mid ((upper_volume (f,D3)),2,(len (upper_volume (f,D3)))) = (upper_volume (f,D3)) /^ 1 by A625, FINSEQ_6:117;
len (mid ((upper_volume (f,D3)),2,(len (upper_volume (f,D3))))) = len (upper_volume (f,D)) by A591, A600, INTEGRA1:def_6;
then A629: (upper_volume (f,D3)) /^ 1 = upper_volume (f,D) by A628, A601, 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 A583, INTEGRA1:def_4
.= (D3 . 1) - (lower_bound A) by A583, INTEGRA1:def_4
.= 0 by A585 ;
then y = 0 + (Sum (upper_volume (f,D))) by A588, A624, A586, A629, RVSUM_1:76
.= upper_sum (f,D) by INTEGRA1:def_8 ;
then y = (upper_sum_set f) . D by INTEGRA1:def_10;
hence 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 ) by A585, A627, A626, A595; ::_thesis: verum
end;
hence 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 ) ; ::_thesis: verum
end;
end;
end;
then consider D being Division of A such that
D in dom (upper_sum_set f) and
A630: y = (upper_sum_set f) . D and
A631: D . 1 > lower_bound A ;
deffunc H1( Nat) -> Element of REAL = vol (divset (D,$1));
set p = len D;
set H = upper_bound (rng f);
set h = lower_bound (rng f);
consider v being FinSequence of REAL such that
A632: ( len v = len D & ( for j being Nat st j in dom v holds
v . j = H1(j) ) ) from FINSEQ_2:sch_1();
A633: 2 * (len D) > 0 by XREAL_1:129;
consider v1 being non-decreasing FinSequence of REAL such that
A634: v,v1 are_fiberwise_equipotent by INTEGRA2:3;
defpred S1[ Nat] means ( $1 in dom v1 & v1 . $1 > 0 );
A635: dom v = Seg (len D) by A632, FINSEQ_1:def_3;
A636: ex k being Nat st S1[k]
proof
consider H being Function such that
dom H = dom v and
rng H = dom v1 and
H is one-to-one and
A637: v = v1 * H by A634, CLASSES1:77;
consider k being Element of NAT such that
A638: k in dom D and
A639: vol (divset (D,k)) > 0 by A565, Th2;
A640: dom D = Seg (len D) by FINSEQ_1:def_3;
then H . k in dom v1 by A635, A637, A638, FUNCT_1:11;
then reconsider Hk = H . k as Element of NAT ;
v . k > 0 by A632, A635, A638, A639, A640;
then S1[Hk] by A635, A637, A638, A640, FUNCT_1:11, FUNCT_1:12;
hence ex k being Nat st S1[k] ; ::_thesis: verum
end;
consider k being Nat such that
A641: ( S1[k] & ( for n being Nat st S1[n] holds
k <= n ) ) from NAT_1:sch_5(A636);
A642: (upper_bound (rng f)) - (lower_bound (rng f)) >= 0 by A1, Lm3, XREAL_1:48;
then A643: (2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1) > 0 by A633, XREAL_1:129;
min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) > 0
proof
percases ( min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) = v1 . k or min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) = e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) ) by XXREAL_0:15;
suppose min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) = v1 . k ; ::_thesis: min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) > 0
hence min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) > 0 by A641; ::_thesis: verum
end;
suppose min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) = e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) ; ::_thesis: min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) > 0
hence min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) > 0 by A575, A643, XREAL_1:139; ::_thesis: verum
end;
end;
end;
then consider n being Element of NAT such that
A644: for m being Element of NAT st n <= m holds
( 0 < (delta T) . m & (delta T) . m < min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) ) by A567;
take n ; ::_thesis: for m being Element of NAT st n <= m holds
abs (((upper_sum (f,T)) . m) - (upper_integral f)) < e
A645: y = upper_sum (f,D) by A630, INTEGRA1:def_10;
for m being Element of NAT st n <= m holds
abs (((upper_sum (f,T)) . m) - (upper_integral f)) < e
proof
A646: v1 . 1 > 0
proof
A647: for n1 being Element of NAT st n1 in dom D holds
vol (divset (D,n1)) > 0
proof
let n1 be Element of NAT ; ::_thesis: ( n1 in dom D implies vol (divset (D,n1)) > 0 )
assume A648: n1 in dom D ; ::_thesis: vol (divset (D,n1)) > 0
then A649: 1 <= n1 by FINSEQ_3:25;
percases ( n1 = 1 or n1 > 1 ) by A649, XXREAL_0:1;
supposeA650: n1 = 1 ; ::_thesis: vol (divset (D,n1)) > 0
then A651: upper_bound (divset (D,n1)) = D . n1 by A648, INTEGRA1:def_4;
lower_bound (divset (D,n1)) = lower_bound A by A648, A650, INTEGRA1:def_4;
then vol (divset (D,n1)) = (D . n1) - (lower_bound A) by A651, INTEGRA1:def_5;
hence vol (divset (D,n1)) > 0 by A631, A650, XREAL_1:50; ::_thesis: verum
end;
supposeA652: n1 > 1 ; ::_thesis: vol (divset (D,n1)) > 0
then A653: upper_bound (divset (D,n1)) = D . n1 by A648, INTEGRA1:def_4;
lower_bound (divset (D,n1)) = D . (n1 - 1) by A648, A652, INTEGRA1:def_4;
then A654: vol (divset (D,n1)) = (D . n1) - (D . (n1 - 1)) by A653, INTEGRA1:def_5;
n1 < n1 + 1 by XREAL_1:29;
then A655: n1 - 1 < n1 by XREAL_1:19;
n1 - 1 in dom D by A648, A652, INTEGRA1:7;
then D . (n1 - 1) < D . n1 by A648, A655, SEQM_3:def_1;
hence vol (divset (D,n1)) > 0 by A654, XREAL_1:50; ::_thesis: verum
end;
end;
end;
A656: k <= len v1 by A641, FINSEQ_3:25;
1 <= k by A641, FINSEQ_3:25;
then 1 <= len v1 by A656, XXREAL_0:2;
then 1 in dom v1 by FINSEQ_3:25;
then A657: v1 . 1 in rng v1 by FUNCT_1:def_3;
rng v = rng v1 by A634, CLASSES1:75;
then consider n1 being Element of NAT such that
A658: n1 in dom v and
A659: v1 . 1 = v . n1 by A657, PARTFUN1:3;
n1 in Seg (len D) by A632, A658, FINSEQ_1:def_3;
then A660: n1 in dom D by FINSEQ_1:def_3;
v1 . 1 = vol (divset (D,n1)) by A632, A658, A659;
hence v1 . 1 > 0 by A647, A660; ::_thesis: verum
end;
A661: v1 . k = min (rng (upper_volume ((chi (A,A)),D)))
proof
A662: k = 1
proof
len v1 = len v by A634, RFINSEQ:3;
then k in Seg (len v) by A641, FINSEQ_1:def_3;
then A663: 1 <= k by FINSEQ_1:1;
k in Seg (len v1) by A641, FINSEQ_1:def_3;
then k <= len v1 by FINSEQ_1:1;
then 1 <= len v1 by A663, XXREAL_0:2;
then A664: 1 in dom v1 by FINSEQ_3:25;
assume k <> 1 ; ::_thesis: contradiction
then k > 1 by A663, XXREAL_0:1;
hence contradiction by A641, A646, A664; ::_thesis: verum
end;
A665: rng v = rng v1 by A634, CLASSES1:75;
v1 . k in rng (upper_volume ((chi (A,A)),D))
proof
v1 . k in rng v by A641, A665, FUNCT_1:def_3;
then consider k2 being Element of NAT such that
A666: k2 in dom v and
A667: v1 . k = v . k2 by PARTFUN1:3;
A668: k2 in Seg (len D) by A632, A666, FINSEQ_1:def_3;
then A669: k2 in dom D by FINSEQ_1:def_3;
k2 in Seg (len (upper_volume ((chi (A,A)),D))) by A668, INTEGRA1:def_6;
then A670: k2 in dom (upper_volume ((chi (A,A)),D)) by FINSEQ_1:def_3;
v1 . k = vol (divset (D,k2)) by A632, A666, A667;
then v1 . k = (upper_volume ((chi (A,A)),D)) . k2 by A669, INTEGRA1:20;
hence v1 . k in rng (upper_volume ((chi (A,A)),D)) by A670, FUNCT_1:def_3; ::_thesis: verum
end;
then A671: 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 being Element of NAT such that
A672: m in dom (upper_volume ((chi (A,A)),D)) and
A673: 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 A672, FINSEQ_1:def_3;
then A674: m in Seg (len D) by INTEGRA1:def_6;
then m in dom D by FINSEQ_1:def_3;
then min (rng (upper_volume ((chi (A,A)),D))) = vol (divset (D,m)) by A673, INTEGRA1:20;
then A675: v . m = min (rng (upper_volume ((chi (A,A)),D))) by A632, A635, A674;
m in dom v by A632, A674, FINSEQ_1:def_3;
then min (rng (upper_volume ((chi (A,A)),D))) in rng v by A675, FUNCT_1:def_3;
then consider m1 being Element of NAT such that
A676: m1 in dom v1 and
A677: min (rng (upper_volume ((chi (A,A)),D))) = v1 . m1 by A665, PARTFUN1:3;
m1 >= 1 by A676, FINSEQ_3:25;
then v1 . 1 <= min (rng (upper_volume ((chi (A,A)),D))) by A641, A662, A676, A677, INTEGRA2:2;
hence v1 . k = min (rng (upper_volume ((chi (A,A)),D))) by A662, A671, XXREAL_0:1; ::_thesis: verum
end;
A678: min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) <= v1 . k by XXREAL_0:17;
set s = upper_integral f;
set sD = upper_sum (f,D);
let m be Element of NAT ; ::_thesis: ( n <= m implies abs (((upper_sum (f,T)) . m) - (upper_integral f)) < e )
reconsider D1 = T . m as Division of A ;
A679: delta D1 = (delta T) . m by Def2;
consider D2 being Division of A such that
A680: D <= D2 and
D1 <= D2 and
A681: 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 . m));
set sD2 = upper_sum (f,D2);
upper_sum (f,D2) <= upper_sum (f,D) by A1, A680, INTEGRA1:45;
then A682: (upper_sum (f,(T . m))) - (upper_sum (f,D)) <= (upper_sum (f,(T . m))) - (upper_sum (f,D2)) by XREAL_1:10;
(((upper_sum (f,D)) + (upper_sum (f,(T . m)))) - (upper_sum (f,(T . m)))) - (upper_integral f) < e / 2 by A579, A645, XREAL_1:19;
then (((upper_sum (f,(T . m))) - (upper_integral f)) + (upper_sum (f,D))) - (upper_sum (f,(T . m))) < e / 2 ;
then ((upper_sum (f,(T . m))) - (upper_integral f)) + (upper_sum (f,D)) < (upper_sum (f,(T . m))) + (e / 2) by XREAL_1:19;
then A683: (upper_sum (f,(T . m))) - (upper_integral f) < ((upper_sum (f,(T . m))) + (e / 2)) - (upper_sum (f,D)) by XREAL_1:20;
T . m in divs A by INTEGRA1:def_3;
then A684: T . m in dom (upper_sum_set f) by FUNCT_2:def_1;
(upper_sum (f,T)) . m = upper_sum (f,(T . m)) 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 A684, FUNCT_1:def_3;
then lower_bound (rng (upper_sum_set f)) <= (upper_sum (f,T)) . m by A577, SEQ_4:def_2;
then upper_integral f <= (upper_sum (f,T)) . m by INTEGRA1:def_14;
then A685: ((upper_sum (f,T)) . m) - (upper_integral f) >= 0 by XREAL_1:48;
(upper_bound (rng f)) - (lower_bound (rng f)) <= ((upper_bound (rng f)) - (lower_bound (rng f))) + 1 by XREAL_1:29;
then A686: (len D) * ((upper_bound (rng f)) - (lower_bound (rng f))) <= (len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1) by XREAL_1:64;
A687: min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) <= e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) by XXREAL_0:17;
assume A688: n <= m ; ::_thesis: abs (((upper_sum (f,T)) . m) - (upper_integral f)) < e
then (delta T) . m < min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) by A644;
then (delta T) . m < e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) by A687, XXREAL_0:2;
then ((delta T) . m) * ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) < e by A633, A642, XREAL_1:79, XREAL_1:129;
then (((delta T) . m) * ((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1))) * 2 < e ;
then A689: ((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) * ((delta T) . m) < e / 2 by XREAL_1:81;
(delta T) . m < min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) by A644, A688;
then delta D1 < v1 . k by A679, A678, 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, A661;
then A690: (upper_sum (f,D1)) - (upper_sum (f,D2)) <= ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by A681, Th6;
0 < (delta T) . m by A644, A688;
then ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * ((delta T) . m) <= ((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) * ((delta T) . m) by A686, XREAL_1:64;
then (upper_sum (f,(T . m))) - (upper_sum (f,D2)) <= ((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) * ((delta T) . m) by A679, A690, XXREAL_0:2;
then (upper_sum (f,(T . m))) - (upper_sum (f,D)) <= ((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) * ((delta T) . m) by A682, XXREAL_0:2;
then (upper_sum (f,(T . m))) - (upper_sum (f,D)) < e / 2 by A689, XXREAL_0:2;
then ((upper_sum (f,(T . m))) - (upper_sum (f,D))) + (e / 2) < (e / 2) + (e / 2) by XREAL_1:6;
then (upper_sum (f,(T . m))) - (upper_integral f) < e by A683, XXREAL_0:2;
then ((upper_sum (f,T)) . m) - (upper_integral f) < e by INTEGRA2:def_2;
hence abs (((upper_sum (f,T)) . m) - (upper_integral f)) < e by A685, ABSVALUE:def_1; ::_thesis: verum
end;
hence for m being Element of NAT st n <= m holds
abs (((upper_sum (f,T)) . m) - (upper_integral f)) < e ; ::_thesis: verum
end;
hence upper_sum (f,T) is convergent by SEQ_2:def_6; ::_thesis: lim (upper_sum (f,T)) = upper_integral f
hence lim (upper_sum (f,T)) = upper_integral f by A574, SEQ_2:def_7; ::_thesis: verum
end;