let A be closed-interval Subset of REAL ; :: thesis:
let f be Function of A,REAL ; :: thesis:
let T be DivSequence of A; :: thesis:
assume that
A1: f | A is bounded and
A2: delta T is convergent_to_0 and
A3: vol A <> 0 ; :: thesis:
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:
consider D2 being Division of A such that
A6: D <= D2 and
A7: D1 <= D2 and
A8: rng D2 = (rng D1) \/ (rng D) by Th3;
A9: (lower_sum f,D2) - (lower_sum f,D1) >= 0 by A1, A7, INTEGRA1:48, XREAL_1:50;
(lower_sum f,D2) - (lower_sum f,D) >= 0 by A1, A6, INTEGRA1:48, XREAL_1:50;
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:

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:
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) )

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:
assume A17: i in dom D ; :: thesis:
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:
assume A20: i in dom D ; :: thesis:
assume that
A21: j in dom D1 and
A22: D . i in divset D1,j ; :: thesis:
assume j < 2 ; :: thesis:
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_1:3;
then j = 1 by A23, XXREAL_0:1;
then A24: lower_bound (divset D1,j) = lower_bound A by A21, INTEGRA1:def 5;
A25: D . i <= upper_bound (divset D1,j) by A22, INTEGRA2:1;
delta D1 >= min (rng (upper_volume (chi A,A),D))

proof

per cases ;

suppose A26: i = 1 ; :: thesis:

len D in Seg (len D) by FINSEQ_1:5;
then 1 <= len D by FINSEQ_1:3;
then A27: 1 in Seg (len D) by FINSEQ_1:3;
then A28: 1 in dom D by FINSEQ_1:def 3;
then A29: lower_bound (divset D,1) = lower_bound A by INTEGRA1:def 5;
1 in Seg (len (upper_volume (chi A,A),D)) by A27, INTEGRA1:def 7;
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, INTEGRA1:22;
then vol (divset D,1) in rng (upper_volume (chi A,A),D) by A30, FUNCT_1:def 5;
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 5;
(upper_bound (divset D1,j)) - (lower_bound A) >= (D . 1) - (lower_bound A) by A25, A26, XREAL_1:11;
then vol (divset D1,j) >= (upper_bound (divset D,1)) - (lower_bound (divset D,1)) by A24, A29, A32, INTEGRA1:def 6;
then A33: vol (divset D1,j) >= vol (divset D,1) by INTEGRA1:def 6;
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:

end;

suppose A34: i <> 1 ; :: thesis:

then D . (i - 1) in A by A20, INTEGRA1:9;
then A35: lower_bound A <= D . (i - 1) by INTEGRA2:1;
lower_bound (divset D,i) = D . (i - 1) by A20, A34, INTEGRA1:def 5;
then A36: (upper_bound (divset D,i)) - (lower_bound A) >= (upper_bound (divset D,i)) - (lower_bound (divset D,i)) by A35, XREAL_1:12;
upper_bound (divset D,i) = D . i by A20, A34, INTEGRA1:def 5;
then (upper_bound (divset D1,j)) - (lower_bound (divset D1,j)) >= (upper_bound (divset D,i)) - (lower_bound A) by A25, A24, XREAL_1:11;
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 6;
then A37: vol (divset D1,j) >= vol (divset D,i) by INTEGRA1:def 6;
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 7;
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, INTEGRA1:22;
then vol (divset D,i) in rng (upper_volume (chi A,A),D) by A38, FUNCT_1:def 5;
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:

end;

hence contradiction by A11; :: thesis:

end;

A40: S₁[1]

proof

len D in Seg (len D) by FINSEQ_1:5;
then 1 <= len D by FINSEQ_1:3;
then A41: 1 in dom D by FINSEQ_3:27;
then consider j being Element of NAT such that
A42: j in dom D1 and
A43: D . 1 in divset D1,j by Th2, INTEGRA1:8;
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, INTEGRA1:9;
A45: j1 in dom D1 by A42, A44, INTEGRA1:9;
then j1 in Seg (len D1) by FINSEQ_1:def 3;
then j1 in Seg (len (lower_volume f,D1)) by INTEGRA1:def 8;
then A46: j1 in dom (lower_volume f,D1) by FINSEQ_1:def 3;
A47: j - 1 in dom D1 by A42, A44, INTEGRA1:9;
then A48: indx D2,D1,j1 in dom D2 by A13, INTEGRA1:def 21;
then A49: indx D2,D1,j1 in Seg (len D2) by FINSEQ_1:def 3;
then A50: 1 <= indx D2,D1,j1 by FINSEQ_1:3;
then mid D2,1,(indx D2,D1,j1) is increasing by A48, INTEGRA1:37;
then A51: D2 | (indx D2,D1,j1) is increasing by A50, FINSEQ_6:122;
j < j + 1 by NAT_1:13;
then j1 < j by XREAL_1:21;
then A52: indx D2,D1,j1 < indx D2,D1,j by A13, A42, A45, Th7;
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:
then A56: (indx D2,D1,j1) + (1 + 1) < indx D2,D1,j by XREAL_1:22;
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 21;
then A60: indx D2,D1,j <= len D2 by FINSEQ_3:27;
then ID2 <= len D2 by A56, XXREAL_0:2;
then ID2 in Seg (len D2) by A58, FINSEQ_1:3;
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 21;
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_1:3;
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 21;
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:

per cases by A70;

suppose D2 . ID1 in rng D1 ; :: thesis:

then consider n being Element of NAT such that
A71: n in dom D1 and
A72: D1 . n = D2 . ID1 by PARTFUN1:26;
j1 < n by A45, A67, A68, A71, A72, SEQ_4:154;
then A73: j < n + 1 by XREAL_1:21;
D2 . ID1 < D2 . (indx D2,D1,j) by A66, A62, XXREAL_0:2;
then n < j by A42, A64, A71, A72, SEQ_4:154;
hence contradiction by A73, NAT_1:13; :: thesis:

end;

suppose D2 . ID2 in rng D1 ; :: thesis:

then consider n being Element of NAT such that
A74: n in dom D1 and
A75: D1 . n = D2 . ID2 by PARTFUN1:26;
D2 . (indx D2,D1,j1) < D2 . ID2 by A67, A66, XXREAL_0:2;
then j1 < n by A45, A68, A74, A75, SEQ_4:154;
then A76: j < n + 1 by XREAL_1:21;
n < j by A42, A62, A64, A74, A75, SEQ_4:154;
hence contradiction by A76, NAT_1:13; :: thesis:

end;

upper_bound (divset D1,j) = D1 . j by A42, A44, INTEGRA1:def 5;
then A77: upper_bound (divset D1,j) = D2 . (indx D2,D1,j) by A13, A42, INTEGRA1:def 21;
lower_bound (divset D1,j) = D1 . j1 by A42, A44, INTEGRA1:def 5;
then A78: lower_bound (divset D1,j) = D2 . (indx D2,D1,j1) by A13, A45, INTEGRA1:def 21;
D2 . ID2 in (rng D) \/ (rng D1) by A14, A61, FUNCT_1:def 5;
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 5;
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, Th4, SEQ_4:155; :: thesis:

end;

A82: 1 <= (indx D2,D1,j1) + 1 by A50, NAT_1:13;
j <= len D1 by A42, FINSEQ_3:27;
then A83: j <= len (lower_volume f,D1) by INTEGRA1:def 8;
A84: 1 <= j by A42, FINSEQ_3:27;
then A85: (mid (lower_volume f,D1),j,j) . 1 = (lower_volume f,D1) . j by A83, FINSEQ_6:124;
(j -' j) + 1 = 1 by Lm1;
then len (mid (lower_volume f,D1),j,j) = 1 by A84, A83, FINSEQ_6:124;
then mid (lower_volume f,D1),j,j = <*((lower_volume f,D1) . j)*> by A85, FINSEQ_1:57;
then A86: Sum (mid (lower_volume f,D1),j,j) = (lower_volume f,D1) . j by FINSOP_1:12;
indx D2,D1,j in dom D2 by A13, A42, INTEGRA1:def 21;
then A87: indx D2,D1,j in Seg (len D2) by FINSEQ_1:def 3;
then A88: 1 <= indx D2,D1,j by FINSEQ_1:3;
indx D2,D1,j in Seg (len (lower_volume f,D2)) by A87, INTEGRA1:def 8;
then A89: indx D2,D1,j <= len (lower_volume f,D2) by FINSEQ_1:3;
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_1:3;
then A91: (indx D2,D1,j1) + 1 in Seg (len D2) by INTEGRA1:def 8;
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_1:235;
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_6:124;
((indx D2,D1,j) -' ((indx D2,D1,j1) + 1)) + 1 >= 0 + 1 by XREAL_1:8;
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_6:124;

now

per cases by A94, A93, Lm2;

suppose A95: len (mid (lower_volume f,D2),((indx D2,D1,j1) + 1),(indx D2,D1,j)) = 1 ; :: thesis:

end;

suppose A104: len (mid (lower_volume f,D2),((indx D2,D1,j1) + 1),(indx D2,D1,j)) = 2 ; :: thesis:

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_6:124;
A106: 2 + ((indx D2,D1,j1) + 1) >= 0 + 1 by XREAL_1:9;
(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_6:124
.= H₁(D2) . ((2 + ((indx D2,D1,j1) + 1)) - 1) by A106, XREAL_1:235
.= 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_1:61;
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_1:107;
A108: vol (divset D2,((indx D2,D1,j1) + 1)) >= 0 by INTEGRA1:11;
upper_bound (divset D1,j) = D1 . j by A42, A44, INTEGRA1:def 5;
then A109: upper_bound (divset D1,j) = D2 . (indx D2,D1,j) by A13, A42, INTEGRA1:def 21;
A110: vol (divset D2,((indx D2,D1,j1) + 2)) >= 0 by INTEGRA1:11;
((indx D2,D1,j) -' ((indx D2,D1,j1) + 1)) + 1 = 2 by A53, A88, A89, A82, A90, A104, FINSEQ_6:124;
then A111: ((indx D2,D1,j) - ((indx D2,D1,j1) + 1)) + 1 = 2 by A53, XREAL_1:235;
then A112: (indx D2,D1,j1) + 2 in dom D2 by A13, A42, INTEGRA1:def 21;
lower_bound (divset D1,j) = D1 . j1 by A42, A44, INTEGRA1:def 5;
then lower_bound (divset D1,j) = D2 . (indx D2,D1,j1) by A13, A45, INTEGRA1:def 21;
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 6;
(indx D2,D1,j1) + 1 in Seg (len (lower_volume f,D2)) by A82, A90, FINSEQ_1:3;
then (indx D2,D1,j1) + 1 in Seg (len D2) by INTEGRA1:def 8;
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 5;
((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 5;
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 5;
upper_bound (divset D2,((indx D2,D1,j1) + 2)) = D2 . ((indx D2,D1,j1) + 2) by A112, A118, INTEGRA1:def 5;
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 6
.= (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 6;
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 8;
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, INTEGRA1:10;
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 INTEGRA1:10;
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_1:66;
((indx D2,D1,j) -' ((indx D2,D1,j1) + 1)) + 1 = 2 by A53, A88, A89, A82, A90, A104, FINSEQ_6:124;
then A125: ((indx D2,D1,j) - ((indx D2,D1,j1) + 1)) + 1 = 2 by A53, XREAL_1:235;
A126: indx D2,D1,j in dom D2 by A13, A42, INTEGRA1:def 21;
then divset D2,((indx D2,D1,j1) + 2) c= A by A125, INTEGRA1:10;
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_1:66;
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_1:21;
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_1:21;
(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_1:66;
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_1:21;
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 8
.= ((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 8 ;
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_1:66;
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_1:22;
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_1:22;
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_1:22;
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_1:15;
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:

end;

(upper_bound (rng f)) - (lower_bound (rng f)) >= 0 by A1, Lm3, XREAL_1:50;
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_1:66;
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:

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:

end;

j < j + 1 by NAT_1:13;
then A131: j1 < j by XREAL_1:21;
indx D2,D1,j in dom D2 by A13, A42, INTEGRA1:def 21;
then A132: indx D2,D1,j in Seg (len D2) by FINSEQ_1:def 3;
then A133: 1 <= indx D2,D1,j by FINSEQ_1:3;
A134: indx D2,D1,j1 <= len D2 by A49, FINSEQ_1:3;
then A135: len (D2 | (indx D2,D1,j1)) = indx D2,D1,j1 by FINSEQ_1:80;
A136: j1 in Seg (len D1) by A47, FINSEQ_1:def 3;
then A137: j1 <= len D1 by FINSEQ_1:3;
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:
assume x1 in rng (D1 | j1) ; :: thesis:
then consider k being Element of NAT such that
A138: k in dom (D1 | j1) and
A139: x1 = (D1 | j1) . k by PARTFUN1:26;
k in Seg (len (D1 | j1)) by A138, FINSEQ_1:def 3;
then A140: k in Seg j1 by A137, FINSEQ_1:80;
then A141: k in dom D1 by A45, RFINSEQ:19;
k <= j1 by A140, FINSEQ_1:3;
then D1 . k <= D1 . j1 by A47, A141, SEQ_4:154;
then D2 . (indx D2,D1,k) <= D1 . j1 by A13, A141, INTEGRA1:def 21;
then A142: D2 . (indx D2,D1,k) <= D2 . (indx D2,D1,j1) by A13, A47, INTEGRA1:def 21;
A143: (D1 | j1) . k = D1 . k by A45, A140, RFINSEQ:19;
D1 . k in rng D1 by A141, FUNCT_1:def 5;
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 PARTFUN1:26;
D2 . (indx D2,D1,k) = D2 . n by A13, A139, A143, A141, A145, INTEGRA1:def 21;
then A146: n <= indx D2,D1,j1 by A48, A144, A142, SEQM_3:def 1;
1 <= n by A144, FINSEQ_3:27;
then A147: n in Seg (indx D2,D1,j1) by A146, FINSEQ_1:3;
then n in Seg (len (D2 | (indx D2,D1,j1))) by A134, FINSEQ_1:80;
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:19;
hence x1 in rng (D2 | (indx D2,D1,j1)) by A145, A148, FUNCT_1:def 5; :: thesis:

end;

then A149: rng (D1 | j1) c= rng (D2 | (indx D2,D1,j1)) by TARSKI:def 3;
A150: 1 <= j1 by A136, FINSEQ_1:3;
lower_bound (divset D1,j) <= D . 1 by A43, INTEGRA2:1;
then A151: D1 . j1 <= D . 1 by A42, A44, INTEGRA1:def 5;
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:
assume x1 in rng (D2 | (indx D2,D1,j1)) ; :: thesis:
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 PARTFUN1:26;
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_1:80;
then A155: k in dom D2 by A48, RFINSEQ:19;
A156: len (D1 | j1) = j1 by A137, FINSEQ_1:80;
k <= indx D2,D1,j1 by A154, FINSEQ_1:3;
then D2 . k <= D2 . (indx D2,D1,j1) by A48, A155, SEQ_4:154;
then A157: D2 . k <= D1 . j1 by A13, A47, INTEGRA1:def 21;
A158: ( D2 . k in rng D1 implies D2 . k in rng (D1 | j1) )

proof

assume D2 . k in rng D1 ; :: thesis:
then consider m being Element of NAT such that
A159: m in dom D1 and
A160: D2 . k = D1 . m by PARTFUN1:26;
m in Seg (len D1) by A159, FINSEQ_1:def 3;
then A161: 1 <= m by FINSEQ_1:3;
A162: m <= j1 by A45, A157, A159, A160, SEQM_3:def 1;
then m in Seg j1 by A161, FINSEQ_1:3;
then A163: D2 . k = (D1 | j1) . m by A45, A160, RFINSEQ:19;
m in dom (D1 | j1) by A156, A161, A162, FINSEQ_3:27;
hence D2 . k in rng (D1 | j1) by A163, FUNCT_1:def 5; :: thesis:

end;

A164: ( D2 . k in rng D implies D2 . k = D1 . j1 )

proof

assume D2 . k in rng D ; :: thesis:
then consider n being Element of NAT such that
A165: n in dom D and
A166: D2 . k = D . n by PARTFUN1:26;
1 <= n by A165, FINSEQ_3:27;
then D . 1 <= D2 . k by A41, A165, A166, SEQ_4:154;
then D1 . j1 <= D2 . k by A151, XXREAL_0:2;
hence D2 . k = D1 . j1 by A157, XXREAL_0:1; :: thesis:

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_1:3;
then j1 in dom (D1 | j1) by FINSEQ_1:def 3;
then A168: (D1 | j1) . j1 in rng (D1 | j1) by FUNCT_1:def 5;
assume A169: D2 . k in rng D ; :: thesis:
j1 in Seg j1 by A150, FINSEQ_1:3;
hence D2 . k in rng (D1 | j1) by A45, A164, A169, A168, RFINSEQ:19; :: thesis:

end;

D2 . k in rng D2 by A155, FUNCT_1:def 5;
hence x1 in rng (D1 | j1) by A14, A48, A153, A154, A167, A158, RFINSEQ:19, XBOOLE_0:def 3; :: thesis:

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, INTEGRA1:9, INTEGRA1:37;
then A171: D1 | j1 is increasing by A150, FINSEQ_6:122;
then A172: D2 | (indx D2,D1,j1) = D1 | j1 by A51, A170, Th5;
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:
assume that
A174: 1 <= k and
A175: k <= j1 ; :: thesis:
assume A176: k <> indx D2,D1,k ; :: thesis:

now

per cases by A176, XXREAL_0:1;

suppose A177: k > indx D2,D1,k ; :: thesis:

k <= len D1 by A137, A175, XXREAL_0:2;
then A178: k in dom D1 by A174, FINSEQ_3:27;
then indx D2,D1,k in dom D2 by A13, INTEGRA1:def 21;
then indx D2,D1,k in Seg (len D2) by FINSEQ_1:def 3;
then A179: 1 <= indx D2,D1,k by FINSEQ_1:3;
A180: indx D2,D1,k < j1 by A175, A177, XXREAL_0:2;
then A181: indx D2,D1,k in Seg j1 by A179, FINSEQ_1:3;
indx D2,D1,k <= indx D2,D1,j1 by A13, A45, A175, A178, Th6;
then indx D2,D1,k in Seg (indx D2,D1,j1) by A179, FINSEQ_1:3;
then A182: (D2 | (indx D2,D1,j1)) . (indx D2,D1,k) = D2 . (indx D2,D1,k) by A48, RFINSEQ:19;
indx D2,D1,k <= len D1 by A137, A180, XXREAL_0:2;
then indx D2,D1,k in dom D1 by A179, FINSEQ_3:27;
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 21;
hence contradiction by A45, A172, A182, A183, A181, RFINSEQ:19; :: thesis:

end;

suppose A184: k < indx D2,D1,k ; :: thesis:

k <= len D1 by A137, A175, XXREAL_0:2;
then A185: k in dom D1 by A174, FINSEQ_3:27;
then indx D2,D1,k <= indx D2,D1,j1 by A13, A45, A175, Th6;
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_3:27;
k in Seg j1 by A174, A175, FINSEQ_1:3;
then A188: D1 . k = (D1 | j1) . k by A47, RFINSEQ:19;
indx D2,D1,k in dom D2 by A13, A185, INTEGRA1:def 21;
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_1:3;
D1 . k = D2 . (indx D2,D1,k) by A13, A185, INTEGRA1:def 21;
hence contradiction by A48, A172, A188, A189, A190, RFINSEQ:19; :: thesis:

end;

hence contradiction ; :: thesis:

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 8;
then A192: indx D2,D1,j1 in dom (lower_volume f,D2) by FINSEQ_1:def 3;
let k be Nat; :: thesis:
assume that
A193: 1 <= k and
A194: k <= len ((lower_volume f,D1) | j1) ; :: thesis:
reconsider k = k as Element of NAT by ORDINAL1:def 13;
A195: len (lower_volume f,D1) = len D1 by INTEGRA1:def 8;
then A196: k <= j1 by A137, A194, FINSEQ_1:80;
then k <= len D1 by A137, XXREAL_0:2;
then A197: k in Seg (len D1) by A193, FINSEQ_1:3;
then A198: k in dom D1 by FINSEQ_1:def 3;
then A199: indx D2,D1,k in dom D2 by A13, INTEGRA1:def 21;
A200: k in Seg j1 by A193, A196, FINSEQ_1:3;
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_1:3;
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 21;
A204: ( lower_bound (divset D1,k) = lower_bound (divset D2,(indx D2,D1,k)) & upper_bound (divset D1,k) = upper_bound (divset D2,(indx D2,D1,k)) )

proof

per cases ;

suppose A205: k = 1 ; :: thesis:

then A206: upper_bound (divset D1,k) = D1 . k by A198, INTEGRA1:def 5;
A207: lower_bound (divset D1,k) = lower_bound A by A198, A205, INTEGRA1:def 5;
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 5; :: thesis:

end;

suppose A208: k <> 1 ; :: thesis:

then reconsider k1 = k - 1 as Element of NAT by A198, INTEGRA1:9;
k <= k + 1 by NAT_1:11;
then k1 <= k by XREAL_1:22;
then A209: k1 <= j1 by A196, XXREAL_0:2;
A210: k - 1 in dom D1 by A198, A208, INTEGRA1:9;
then k1 in Seg (len D1) by FINSEQ_1:def 3;
then 1 <= k1 by FINSEQ_1:3;
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 5;
A214: upper_bound (divset D2,(indx D2,D1,k)) = D2 . (indx D2,D1,k) by A199, A212, INTEGRA1:def 5;
A215: upper_bound (divset D1,k) = D1 . k by A198, A208, INTEGRA1:def 5;
lower_bound (divset D1,k) = D1 . (k - 1) by A198, A208, INTEGRA1:def 5;
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 21; :: thesis:

end;

end;

indx D2,D1,j1 in dom D2 by A13, A47, INTEGRA1:def 21;
then indx D2,D1,j1 <= len D2 by FINSEQ_3:27;
then A220: indx D2,D1,j1 <= len (lower_volume f,D2) by INTEGRA1:def 8;
j1 <= len D1 by A47, FINSEQ_3:27;
then A221: j1 <= len (lower_volume f,D1) by INTEGRA1:def 8;
len (D2 | (indx D2,D1,j1)) = len (D1 | j1) by A51, A171, A170, Th5;
then indx D2,D1,j1 = j1 by A137, A135, FINSEQ_1:80;
then len ((lower_volume f,D1) | j1) = indx D2,D1,j1 by A221, FINSEQ_1:80;
then len ((lower_volume f,D1) | j1) = len ((lower_volume f,D2) | (indx D2,D1,j1)) by A220, FINSEQ_1:80;
then A222: (lower_volume f,D2) | (indx D2,D1,j1) = (lower_volume f,D1) | j1 by A191, FINSEQ_1:18;
A223: j in Seg (len D1) by A42, FINSEQ_1:def 3;
then A224: 1 <= j by FINSEQ_1:3;
indx D2,D1,j in Seg (len H₁(D2)) by A132, INTEGRA1:def 8;
then A225: indx D2,D1,j in dom H₁(D2) by FINSEQ_1:def 3;
indx D2,D1,j <= len D2 by A132, FINSEQ_1:3;
then A226: indx D2,D1,j <= len H₁(D2) by INTEGRA1:def 8;
j in Seg (len H₁(D1)) by A223, INTEGRA1:def 8;
then A227: j in dom H₁(D1) by FINSEQ_1:def 3;
j <= len D1 by A223, FINSEQ_1:3;
then A228: j <= len H₁(D1) by INTEGRA1:def 8;
j1 in Seg (len D1) by A45, FINSEQ_1:def 3;
then j1 in Seg (len H₁(D1)) by INTEGRA1:def 8;
then j1 in dom H₁(D1) by FINSEQ_1:def 3;
then H₂(D1,j1) = Sum (H₁(D1) | j1) by INTEGRA1:def 22;
then H₂(D1,j1) + (Sum (mid H₁(D1),j,j)) = Sum ((H₁(D1) | j1) ^ (mid H₁(D1),j,j)) by RVSUM_1:105
.= Sum ((mid H₁(D1),1,j1) ^ (mid H₁(D1),(j1 + 1),j)) by A150, FINSEQ_6:122
.= Sum (mid H₁(D1),1,j) by A150, A228, A131, INTEGRA2:4
.= Sum (H₁(D1) | j) by A224, FINSEQ_6:122 ;
then A229: H₂(D1,j1) + (Sum (mid (lower_volume f,D1),j,j)) = H₂(D1,j) by A227, INTEGRA1:def 22;
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 8;
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 22;
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_1:105
.= Sum ((mid H₁(D2),1,(indx D2,D1,j1)) ^ (mid H₁(D2),((indx D2,D1,j1) + 1),(indx D2,D1,j))) by A50, FINSEQ_6:122
.= 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_6:122 ;
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 22;
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 8;
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 22
.= H₂(D1,j1) by A222, A46, INTEGRA1:def 22 ;
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:

end;

hence S₁[1] by A42, A43; :: thesis:

end;

reconsider i = i as non empty Element of NAT by A18, FINSEQ_1:3;
A231: for i being non empty Nat st S₁[i] holds
S₁[i + 1]

proof

let i be non empty Nat; :: thesis:
A232: i >= 1 by NAT_1:14;
assume A233: S₁[i] ; :: thesis:
S₁[i + 1]

proof

A234: i <= i + 1 by NAT_1:11;
assume A235: i + 1 in dom D ; :: thesis:
then consider j being Element of NAT such that
A236: j in dom D1 and
A237: D . (i + 1) in divset D1,j by Th2, INTEGRA1:8;
A238: D2 . (indx D2,D1,j) = D1 . j by A13, A236, INTEGRA1:def 21;
i + 1 <= len D by A235, FINSEQ_3:27;
then i <= len D by A234, XXREAL_0:2;
then A239: i in Seg (len D) by A232, FINSEQ_1:3;
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:

now

per cases by A246, XXREAL_0:1;

suppose A247: n1 = j ; :: thesis:

D . i in rng D by A240, FUNCT_1:def 5;
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 5;
then A249: D . (i + 1) in (rng D) /\ (divset D1,j) by A237, XBOOLE_0:def 4;
i + 1 > i by XREAL_1:31;
hence contradiction by A11, A235, A236, A240, A248, A249, Th4, SEQ_4:155; :: thesis:

end;

suppose n1 > j ; :: thesis:

then A250: n1 >= j + 1 by NAT_1:13;
then A251: n1 - 1 >= j by XREAL_1:21;
1 <= j by A236, FINSEQ_3:27;
then 1 + 1 <= j + 1 by XREAL_1:8;
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 5;
n1 - 1 in dom D1 by A241, A252, INTEGRA1:9;
then D1 . j <= D1 . (n1 - 1) by A236, A251, SEQ_4:154;
then A254: D . i >= D1 . j by A253, XXREAL_0:2;
A255: i < i + 1 by XREAL_1:31;
A256: upper_bound (divset D1,j) = D1 . j

proof

per cases ;

suppose j = 1 ; :: thesis:

hence upper_bound (divset D1,j) = D1 . j by A236, INTEGRA1:def 5; :: thesis:

end;

suppose j <> 1 ; :: thesis:

hence upper_bound (divset D1,j) = D1 . j by A236, INTEGRA1:def 5; :: thesis:

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:

end;

hence contradiction ; :: thesis:

end;

then A257: n1 + 1 <= j by NAT_1:13;
A258: 1 <= n1 by A241, FINSEQ_3:27;
A259: indx D2,D1,n1 in dom D2 by A13, A241, INTEGRA1:def 21;
then A260: 1 <= indx D2,D1,n1 by FINSEQ_3:27;
A261: indx D2,D1,j in dom D2 by A13, A236, INTEGRA1:def 21;
then A262: 1 <= indx D2,D1,j by FINSEQ_3:27;
A263: indx D2,D1,j <= len D2 by A261, FINSEQ_3:27;
then A264: indx D2,D1,j <= len H₁(D2) by INTEGRA1:def 8;
A265: 1 <= j by A236, FINSEQ_3:27;
A266: j <= len D1 by A236, FINSEQ_3:27;
then A267: n1 + 1 <= len D1 by A257, XXREAL_0:2;
then A268: n1 + 1 in dom D1 by A244, FINSEQ_3:27;
then A269: indx D2,D1,(n1 + 1) in dom D2 by A13, INTEGRA1:def 21;
then A270: 1 <= indx D2,D1,(n1 + 1) by FINSEQ_3:27;
A271: D2 . (indx D2,D1,(n1 + 1)) = D1 . (n1 + 1) by A13, A268, INTEGRA1:def 21;
then D2 . (indx D2,D1,(n1 + 1)) <= D2 . (indx D2,D1,j) by A236, A257, A268, A238, SEQ_4:154;
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_1:8;
then 1 <= ((indx D2,D1,j) + 1) - (indx D2,D1,(n1 + 1)) by XREAL_1:21;
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_6:128
.= D1 . (n1 + 1) by A13, A268, INTEGRA1:def 21 ;
A274: D2 . (indx D2,D1,n1) = D1 . n1 by A13, A241, INTEGRA1:def 21;
A275: j <= len H₁(D1) by A266, INTEGRA1:def 8;
then j in Seg (len H₁(D1)) by A265, FINSEQ_1:3;
then A276: j in dom H₁(D1) by FINSEQ_1:def 3;
A277: indx D2,D1,(n1 + 1) <= len D2 by A269, FINSEQ_3:27;
n1 in Seg (len D1) by A241, FINSEQ_1:def 3;
then n1 in Seg (len H₁(D1)) by INTEGRA1:def 8;
then n1 in dom H₁(D1) by FINSEQ_1:def 3;
then H₂(D1,n1) = Sum (H₁(D1) | n1) by INTEGRA1:def 22
.= Sum (mid H₁(D1),1,n1) by A258, FINSEQ_6:122 ;
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_1:105
.= Sum (mid H₁(D1),1,j) by A245, A258, A275, INTEGRA2:4
.= Sum (H₁(D1) | j) by A265, FINSEQ_6:122 ;
then A278: H₂(D1,j) = H₂(D1,n1) + (Sum (mid H₁(D1),(n1 + 1),j)) by A276, INTEGRA1:def 22;
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 8;
then A279: indx D2,D1,j in dom H₁(D2) by FINSEQ_1:def 3;
A280: n1 >= 1 by A241, FINSEQ_3:27;
A281: j - n1 >= 1 by A257, XREAL_1:21;
(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

per cases by A257, XXREAL_0:1;

suppose A282: n1 + 1 = j ; :: thesis:

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 5;
A285: lower_bound (divset D1,j) = D1 . (j - 1) by A236, A245, A280, INTEGRA1:def 5;
A286: 1 <= (indx D2,D1,n1) + 1 by A260, NAT_1:13;
assume (indx D2,D1,j) - (indx D2,D1,n1) > 2 ; :: thesis:
then A287: (indx D2,D1,n1) + 2 < indx D2,D1,j by XREAL_1:22;
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_1:8;
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_3:27;
then A292: D2 . (indx D2,D1,j) >= D2 . ((indx D2,D1,n1) + 2) by A261, A287, SEQ_4:154;
A293: not D2 . ((indx D2,D1,n1) + 2) in rng D1

proof

assume D2 . ((indx D2,D1,n1) + 2) in rng D1 ; :: thesis:
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 PARTFUN1:26;
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_4:154;
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_4:154;
hence contradiction by A282, A296, NAT_1:13; :: thesis:

end;

D2 . ((indx D2,D1,n1) + 2) in rng D2 by A291, FUNCT_1:def 5;
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 5;
A299: upper_bound (divset D1,j) = D1 . j by A236, A245, A280, INTEGRA1:def 5;
D2 . ((indx D2,D1,n1) + 2) >= D2 . (indx D2,D1,n1) by A259, A290, A291, SEQ_4:154;
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_3:27;
then A303: D2 . (indx D2,D1,j) >= D2 . ((indx D2,D1,n1) + 1) by A261, A301, SEQ_4:154;
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:
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 PARTFUN1:26;
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_4:154;
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_4:154;
hence contradiction by A282, A308, NAT_1:13; :: thesis:

end;

D2 . ((indx D2,D1,n1) + 1) in rng D2 by A302, FUNCT_1:def 5;
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_4:154;
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, Th4;
hence contradiction by A289, A302, A291, SEQM_3:def 1; :: thesis:

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:
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_1:22;
hence (indx D2,D1,n1) + 2 = indx D2,D1,j by A311, XXREAL_0:1; :: thesis:

end;

A312: 1 <= (indx D2,D1,n1) + 1 by NAT_1:12;
A313: indx D2,D1,j <= len H₁(D2) by A263, INTEGRA1:def 8;
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_4:154;
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 8;
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_6:124
.= ((indx D2,D1,j) - ((indx D2,D1,n1) + 1)) + 1 by A315, XREAL_1:235
.= (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

per cases ) - (indx D2,D1,n1) = 1 or (indx D2,D1,j) - (indx D2,D1,n1) = 2 ) by A317, A310;

suppose A319: (indx D2,D1,j) - (indx D2,D1,n1) = 1 ; :: thesis:

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 5;
then A321: upper_bound (divset D2,((indx D2,D1,n1) + 1)) = D1 . j by A13, A236, A319, INTEGRA1:def 21;
lower_bound (divset D2,((indx D2,D1,n1) + 1)) = D2 . (((indx D2,D1,n1) + 1) - 1) by A261, A319, A320, INTEGRA1:def 5;
then A322: lower_bound (divset D2,((indx D2,D1,n1) + 1)) = D1 . n1 by A13, A241, INTEGRA1:def 21;
lower_bound (divset D1,(n1 + 1)) = D1 . ((n1 + 1) - 1) by A245, A280, A268, A282, INTEGRA1:def 5;
then A323: divset D2,((indx D2,D1,n1) + 1) = divset D1,(n1 + 1) by A245, A280, A268, A282, A322, A321, INTEGRA1:def 5;
A324: vol (divset D2,((indx D2,D1,n1) + 1)) >= 0 by INTEGRA1:11;
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_6:128
.= 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_1:57;
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 8;
then H₁(D2) . ((indx D2,D1,n1) + 1) <= (upper_bound (rng f)) * (vol (divset D1,(n1 + 1))) by A1, A261, A319, A323, A324, Th17, XREAL_1:66;
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_1:12; :: thesis:

end;

suppose A326: (indx D2,D1,j) - (indx D2,D1,n1) = 2 ; :: thesis:

(indx D2,D1,n1) + 2 >= 2 + 1 by A260, XREAL_1:8;
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 5;
((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 5;
then A329: vol (divset D2,((indx D2,D1,n1) + 2)) = (D1 . j) - (D2 . ((indx D2,D1,n1) + 1)) by A238, A328, INTEGRA1:def 6;
A330: upper_bound (divset D1,(n1 + 1)) = D1 . (n1 + 1) by A245, A280, A268, A282, INTEGRA1:def 5;
lower_bound (divset D1,(n1 + 1)) = D1 . ((n1 + 1) - 1) by A245, A280, A268, A282, INTEGRA1:def 5;
then A331: vol (divset D1,(n1 + 1)) = (D1 . (n1 + 1)) - (D1 . n1) by A330, INTEGRA1:def 6;
A332: vol (divset D2,((indx D2,D1,n1) + 2)) >= 0 by INTEGRA1:11;
A333: indx D2,D1,j <= len H₁(D2) by A263, INTEGRA1:def 8;
A334: vol (divset D2,((indx D2,D1,n1) + 1)) >= 0 by INTEGRA1:11;
A335: 1 <= (indx D2,D1,n1) + 1 by NAT_1:12;
A336: (indx D2,D1,n1) + 1 <= (indx D2,D1,n1) + 2 by XREAL_1:8;
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_3:27;
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 8;
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, Th17, XREAL_1:66;
((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_6:128
.= 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_6:128
.= 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_1:61;
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_1:107;
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 5;
lower_bound (divset D2,((indx D2,D1,n1) + 1)) = D2 . (((indx D2,D1,n1) + 1) - 1) by A337, A341, INTEGRA1:def 5;
then A343: vol (divset D2,((indx D2,D1,n1) + 1)) = (D2 . ((indx D2,D1,n1) + 1)) - (D1 . n1) by A274, A342, INTEGRA1:def 6;
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 8;
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, Th17, XREAL_1:66;
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_1:9;
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:

end;

end;

suppose A352: n1 + 1 < j ; :: thesis:

A353: n1 < n1 + 1 by NAT_1:13;
then A354: D1 . n1 < D1 . (n1 + 1) by A241, A268, SEQM_3:def 1;
then consider B being closed-interval Subset of REAL , MD1, MD2 being Division of B such that
A355: D1 . n1 = lower_bound B and
upper_bound B = MD2 . (len MD2) and
A356: upper_bound B = MD1 . (len MD1) and
A357: MD1 <= MD2 and
A358: MD1 = mid D1,(n1 + 1),j and
A359: MD2 = mid D2,(indx D2,D1,(n1 + 1)),(indx D2,D1,j) by A13, A236, A257, A268, A273, Th14;
A360: delta MD1 >= 0 by Th8;
A361: len MD1 = (j -' (n1 + 1)) + 1 by A257, A265, A266, A244, A267, A358, FINSEQ_6:124;
then A362: ((len MD1) + (n1 + 1)) - 1 = (((j - (n1 + 1)) + 1) + (n1 + 1)) - 1 by A257, XREAL_1:235
.= j ;
j -' (n1 + 1) = j - (n1 + 1) by A257, XREAL_1:235;
then A363: (j -' (n1 + 1)) + 1 = j - n1 ;
then A364: len MD1 = j - n1 by A257, A265, A266, A244, A267, A358, FINSEQ_6:124;
A365: B c= A

proof

let x1 be set ; :: according to TARSKI:def 3 :: thesis:
A366: rng D1 c= A by INTEGRA1:def 2;
D1 . n1 in rng D1 by A241, FUNCT_1:def 5;
then A367: lower_bound A <= D1 . n1 by A366, INTEGRA2:1;
assume A368: x1 in B ; :: thesis:
then reconsider x1 = x1 as Real ;
A369: x1 <= MD1 . (len MD1) by A356, A368, INTEGRA2:1;
D1 . j in rng D1 by A236, FUNCT_1:def 5;
then A370: D1 . j <= upper_bound A by A366, INTEGRA2:1;
D1 . n1 <= x1 by A355, A368, INTEGRA2:1;
then A371: lower_bound A <= x1 by A367, XXREAL_0:2;
MD1 . (len MD1) = D1 . (((j - n1) - 1) + (n1 + 1)) by A257, A281, A266, A244, A358, A363, A364, FINSEQ_6:128
.= D1 . j ;
then x1 <= upper_bound A by A369, A370, XXREAL_0:2;
hence x1 in A by A371, INTEGRA2:1; :: thesis:

end;

then reconsider g = f | B as Function of B,REAL by FUNCT_2:38;
A372: len (lower_volume g,MD1) = len MD1 by INTEGRA1:def 8
.= (j -' (n1 + 1)) + 1 by A257, A265, A266, A244, A267, A358, FINSEQ_6:124
.= (j - (n1 + 1)) + 1 by A257, XREAL_1:235 ;
A373: len MD1 in dom MD1 by FINSEQ_5:6;
then A374: 1 <= len MD1 by FINSEQ_3:27;
A375: ( lower_bound (divset MD1,(len MD1)) = lower_bound (divset D1,j) & upper_bound (divset MD1,(len MD1)) = upper_bound (divset D1,j) )

proof

per cases ;

suppose A376: len MD1 = 1 ; :: thesis:

then A377: upper_bound (divset MD1,(len MD1)) = MD1 . (len MD1) by A373, INTEGRA1:def 5;
A378: upper_bound (divset D1,j) = D1 . j by A236, A245, A280, INTEGRA1:def 5;
lower_bound (divset D1,j) = D1 . (j - 1) by A236, A245, A280, INTEGRA1:def 5;
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, A355, A358, A362, A373, A376, A377, A378, FINSEQ_6:124, INTEGRA1:def 5; :: thesis:

end;

suppose A379: len MD1 <> 1 ; :: thesis:

then (len MD1) - 1 in dom MD1 by A373, INTEGRA1:9;
then A380: (len MD1) - 1 >= 1 by FINSEQ_3:27;
len MD1 <= (len MD1) + 1 by NAT_1:11;
then A381: (len MD1) - 1 <= len MD1 by XREAL_1:22;
upper_bound (divset MD1,(len MD1)) = MD1 . (len MD1) by A373, A379, INTEGRA1:def 5;
then A382: upper_bound (divset MD1,(len MD1)) = D1 . j by A257, A266, A244, A358, A361, A362, A374, FINSEQ_6:128;
A383: (((len MD1) - 1) + (n1 + 1)) - 1 = j - 1 by A364;
lower_bound (divset MD1,(len MD1)) = MD1 . ((len MD1) - 1) by A373, A379, INTEGRA1:def 5;
then lower_bound (divset MD1,(len MD1)) = D1 . (j - 1) by A257, A266, A244, A358, A361, A383, A380, A381, FINSEQ_6:128;
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, A382, INTEGRA1:def 5; :: thesis:

end;

A384: len MD1 in dom MD1 by FINSEQ_5:6;
A385: upper_bound (divset MD1,(len MD1)) = MD1 . (len MD1)

proof

per cases ;

suppose len MD1 = 1 ; :: thesis:

hence upper_bound (divset MD1,(len MD1)) = MD1 . (len MD1) by A384, INTEGRA1:def 5; :: thesis:

end;

suppose len MD1 <> 1 ; :: thesis:

hence upper_bound (divset MD1,(len MD1)) = MD1 . (len MD1) by A384, INTEGRA1:def 5; :: thesis:

end;

D1 . n1 < D1 . (n1 + 1) by A241, A268, A353, SEQM_3:def 1;
then indx D2,D1,n1 < indx D2,D1,(n1 + 1) by A259, A274, A269, A271, SEQ_4:154;
then A386: (indx D2,D1,n1) + 1 <= indx D2,D1,(n1 + 1) by NAT_1:13;
then A387: (indx D2,D1,n1) + 1 <= len D2 by A277, XXREAL_0:2;
vol B = (upper_bound B) - (D1 . n1) by A355, INTEGRA1:def 6;
then vol B = (D1 . j) - (D1 . n1) by A236, A245, A280, A356, A375, A385, INTEGRA1:def 5;
then A388: vol B <> 0 by A236, A241, A245, SEQM_3:def 1;
A389: 1 <= (indx D2,D1,n1) + 1 by A260, NAT_1:13;
A390: indx D2,D1,n1 < (indx D2,D1,n1) + 1 by NAT_1:13;
A391: indx D2,D1,(n1 + 1) = (indx D2,D1,n1) + 1

proof

assume indx D2,D1,(n1 + 1) <> (indx D2,D1,n1) + 1 ; :: thesis:
then A392: indx D2,D1,(n1 + 1) > (indx D2,D1,n1) + 1 by A386, XXREAL_0:1;
A393: (indx D2,D1,n1) + 1 in dom D2 by A389, A387, FINSEQ_3:27;
then A394: D2 . ((indx D2,D1,n1) + 1) in rng D2 by FUNCT_1:def 5;

now

per cases by A14, A394, XBOOLE_0:def 3;

suppose D2 . ((indx D2,D1,n1) + 1) in rng D1 ; :: thesis:

then consider n2 being Element of NAT such that
A395: n2 in dom D1 and
A396: D2 . ((indx D2,D1,n1) + 1) = D1 . n2 by PARTFUN1:26;
D2 . (indx D2,D1,n1) < D2 . ((indx D2,D1,n1) + 1) by A259, A390, A393, SEQM_3:def 1;
then n1 < n2 by A241, A274, A395, A396, SEQ_4:154;
then A397: n1 + 1 <= n2 by NAT_1:13;
D1 . n2 < D1 . (n1 + 1) by A269, A271, A392, A393, A396, SEQM_3:def 1;
hence contradiction by A268, A395, A397, SEQ_4:154; :: thesis:

end;

suppose A398: D2 . ((indx D2,D1,n1) + 1) in rng D ; :: thesis:

A399: D . i <= upper_bound (divset D1,n1) by A242, INTEGRA2:1;
A400: upper_bound (divset D1,n1) = D1 . n1

proof

per cases ;

suppose n1 = 1 ; :: thesis:

hence upper_bound (divset D1,n1) = D1 . n1 by A241, INTEGRA1:def 5; :: thesis:

end;

suppose n1 <> 1 ; :: thesis:

hence upper_bound (divset D1,n1) = D1 . n1 by A241, INTEGRA1:def 5; :: thesis:

end;

consider n2 being Element of NAT such that
A401: n2 in dom D and
A402: D2 . ((indx D2,D1,n1) + 1) = D . n2 by A398, PARTFUN1:26;
D1 . n1 < D . n2 by A259, A274, A390, A393, A402, SEQM_3:def 1;
then D . i < D . n2 by A399, A400, XXREAL_0:2;
then i < n2 by A240, A401, SEQ_4:154;
then A403: i + 1 <= n2 by NAT_1:13;
(n1 + 1) + 1 <= j by A352, NAT_1:13;
then A404: n1 + 1 <= j - 1 by XREAL_1:21;
j - 1 in dom D1 by A236, A245, A280, INTEGRA1:9;
then A405: D1 . (n1 + 1) <= D1 . (j - 1) by A268, A404, SEQ_4:154;
A406: 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 5;
then A407: D1 . (n1 + 1) <= D . (i + 1) by A405, A406, XXREAL_0:2;
D . n2 < D1 . (n1 + 1) by A269, A271, A392, A393, A402, SEQM_3:def 1;
then D . n2 < D . (i + 1) by A407, XXREAL_0:2;
hence contradiction by A235, A401, A403, SEQ_4:154; :: thesis:

end;

hence contradiction ; :: thesis:

end;

A408: j <= len H₁(D1) by A266, INTEGRA1:def 8;
A409: 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:
assume that
A410: 1 <= k and
A411: k <= len (lower_volume g,MD1) ; :: thesis:
A412: k in Seg (len (lower_volume g,MD1)) by A410, A411, FINSEQ_1:3;
then A413: k in Seg (len MD1) by INTEGRA1:def 8;
then A414: k in dom MD1 by FINSEQ_1:def 3;
k in dom MD1 by A413, FINSEQ_1:def 3;
then A415: (lower_volume g,MD1) . k = (lower_bound (rng (g | (divset MD1,k)))) * (vol (divset MD1,k)) by INTEGRA1:def 8;
consider k2 being Element of NAT such that
A416: n1 + 1 = 1 + k2 ;
A417: 1 <= k + k2 by A410, NAT_1:12;
k <= j - ((n1 + 1) - 1) by A372, A411;
then k + ((n1 + 1) - 1) <= j by XREAL_1:21;
then k + k2 <= len D1 by A266, A416, XXREAL_0:2;
then A418: k + k2 in Seg (len D1) by A417, FINSEQ_1:3;
then A419: k + k2 in dom D1 by FINSEQ_1:def 3;
1 + 1 <= k + k2 by A258, A410, A416, XREAL_1:9;
then A420: 1 < k + k2 by NAT_1:13;
A421: k2 = (n1 + 1) - 1 by A416;
A422: ( lower_bound (divset D1,(k + k2)) = lower_bound (divset MD1,k) & upper_bound (divset D1,(k + k2)) = upper_bound (divset MD1,k) )

proof

per cases ;

suppose A423: k = 1 ; :: thesis:

then upper_bound (divset MD1,k) = MD1 . k by A414, INTEGRA1:def 5;
then A424: upper_bound (divset MD1,k) = D1 . ((k + (n1 + 1)) - 1) by A257, A266, A244, A358, A372, A410, A411, A412, FINSEQ_6:128;
lower_bound (divset MD1,k) = D1 . n1 by A355, A414, A423, INTEGRA1:def 5;
hence ( lower_bound (divset D1,(k + k2)) = lower_bound (divset MD1,k) & upper_bound (divset D1,(k + k2)) = upper_bound (divset MD1,k) ) by A421, A420, A419, A423, A424, INTEGRA1:def 5; :: thesis:

end;

suppose A425: k <> 1 ; :: thesis:

then upper_bound (divset MD1,k) = MD1 . k by A414, INTEGRA1:def 5;
then A426: upper_bound (divset MD1,k) = D1 . ((k + (n1 + 1)) - 1) by A257, A266, A244, A358, A372, A410, A411, A412, FINSEQ_6:128;
A427: k - 1 <= (j - (n1 + 1)) + 1 by A372, A411, XREAL_1:148, XXREAL_0:2;
A428: lower_bound (divset MD1,k) = MD1 . (k - 1) by A414, A425, INTEGRA1:def 5;
A429: k - 1 in dom MD1 by A414, A425, INTEGRA1:9;
then 1 <= k - 1 by FINSEQ_3:27;
then lower_bound (divset MD1,k) = D1 . (((k - 1) + (n1 + 1)) - 1) by A257, A266, A244, A358, A429, A427, A428, FINSEQ_6:128;
hence ( lower_bound (divset D1,(k + k2)) = lower_bound (divset MD1,k) & upper_bound (divset D1,(k + k2)) = upper_bound (divset MD1,k) ) by A416, A420, A419, A426, INTEGRA1:def 5; :: thesis:

end;

divset MD1,k = [.(lower_bound (divset MD1,k)),(upper_bound (divset MD1,k)).] by INTEGRA1:5;
then A430: divset D1,(k + k2) = divset MD1,k by A422, INTEGRA1:5;
A431: k + k2 in dom D1 by A418, FINSEQ_1:def 3;
A432: (mid H₁(D1),(n1 + 1),j) . k = H₁(D1) . ((k + (n1 + 1)) - 1) by A257, A244, A372, A408, A410, A411, A412, FINSEQ_6:128
.= (lower_bound (rng (f | (divset D1,(k + k2))))) * (vol (divset D1,(k + k2))) by A416, A431, INTEGRA1:def 8 ;
k in dom MD1 by A413, FINSEQ_1:def 3;
then divset D1,(k + k2) c= B by A430, INTEGRA1:10;
hence (lower_volume g,MD1) . k = (mid H₁(D1),(n1 + 1),j) . k by A415, A432, A430, FUNCT_1:82; :: thesis:

end;

A433: g | B is bounded

proof

consider a being real number such that
A434: for x being set st x in A /\ (dom f) holds
a <= f . x by A1, RFUNCT_1:88;
for x being set st x in B /\ (dom g) holds
a <= g . x

proof

let x be set ; :: thesis:
A435: (dom f) /\ B c= (dom f) /\ A by A365, XBOOLE_1:26;
assume x in B /\ (dom g) ; :: thesis:
then A436: x in dom g by XBOOLE_0:def 4;
then x in (dom f) /\ B by RELAT_1:90;
then a <= f . x by A434, A435;
hence a <= g . x by A436, FUNCT_1:70; :: thesis:

end;

then A437: g | B is bounded_below by RFUNCT_1:88;
consider a being real number such that
A438: for x being set st x in A /\ (dom f) holds
f . x <= a by A1, RFUNCT_1:87;
for x being set st x in B /\ (dom g) holds
g . x <= a

proof

let x be set ; :: thesis:
A439: (dom f) /\ B c= (dom f) /\ A by A365, XBOOLE_1:26;
assume x in B /\ (dom g) ; :: thesis:
then A440: x in dom g by XBOOLE_0:def 4;
then x in (dom f) /\ B by RELAT_1:90;
then a >= f . x by A438, A439;
hence g . x <= a by A440, FUNCT_1:70; :: thesis:

end;

then g | B is bounded_above by RFUNCT_1:87;
hence g | B is bounded by A437; :: thesis:

end;

rng f is bounded_below by A1, INTEGRA1:13;
then A441: lower_bound (rng f) <= lower_bound (rng g) by RELAT_1:99, SEQ_4:64;
rng f is bounded_above by A1, INTEGRA1:15;
then upper_bound (rng f) >= upper_bound (rng g) by RELAT_1:99, SEQ_4:65;
then (upper_bound (rng f)) - (lower_bound (rng f)) >= (upper_bound (rng g)) - (lower_bound (rng g)) by A441, XREAL_1:15;
then A442: ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta MD1) >= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta MD1) by A360, XREAL_1:66;
A443: n1 < j - 1 by A352, XREAL_1:22;
A444: indx D2,D1,j <= len H₁(D2) by A263, INTEGRA1:def 8;
A445: len MD2 = ((indx D2,D1,j) -' (indx D2,D1,(n1 + 1))) + 1 by A272, A270, A277, A262, A263, A359, FINSEQ_6:124;
then A446: len MD2 = ((indx D2,D1,j) - (indx D2,D1,(n1 + 1))) + 1 by A272, XREAL_1:235;
then A447: len (lower_volume g,MD2) = ((indx D2,D1,j) - ((indx D2,D1,n1) + 1)) + 1 by A391, INTEGRA1:def 8;
for x1 being set st x1 in (rng MD1) \/ {(D . (i + 1))} holds
x1 in rng MD2

proof

let x1 be set ; :: thesis:
assume A448: x1 in (rng MD1) \/ {(D . (i + 1))} ; :: thesis:
then reconsider x1 = x1 as Real ;

now

per cases by A448, XBOOLE_0:def 3;

suppose A449: x1 in rng MD1 ; :: thesis:

end;

suppose x1 in {(D . (i + 1))} ; :: thesis:

then A460: x1 = D . (i + 1) by TARSKI:def 1;
reconsider j1 = j - 1 as Element of NAT by A236, A245, A280, INTEGRA1:9;
A461: rng D c= rng D2 by A12, INTEGRA1:def 20;
D . (i + 1) in rng D by A235, FUNCT_1:def 5;
then consider k being Element of NAT such that
A462: k in dom D2 and
A463: x1 = D2 . k by A460, A461, PARTFUN1:26;
D . (i + 1) <= upper_bound (divset D1,j) by A237, INTEGRA2:1;
then x1 <= D1 . j by A236, A245, A280, A460, INTEGRA1:def 5;
then A464: D2 . k <= D2 . (indx D2,D1,j) by A13, A236, A463, INTEGRA1:def 21;
n1 < j1 by A352, XREAL_1:22;
then A465: n1 + 1 <= j1 by NAT_1:13;
j - 1 in dom D1 by A236, A245, A280, INTEGRA1:9;
then A466: D1 . (n1 + 1) <= D1 . (j - 1) by A268, A465, SEQ_4:154;
lower_bound (divset D1,j) <= D . (i + 1) by A237, INTEGRA2:1;
then D1 . (j - 1) <= x1 by A236, A245, A280, A460, INTEGRA1:def 5;
then D2 . (indx D2,D1,(n1 + 1)) <= D2 . k by A271, A463, A466, XXREAL_0:2;
hence x1 in rng MD2 by A269, A261, A359, A462, A463, A464, Th16; :: thesis:

end;

hence x1 in rng MD2 ; :: thesis:

end;

then A467: (rng MD1) \/ {(D . (i + 1))} c= rng MD2 by TARSKI:def 3;
rng MD2 <> {} ;
then 1 in dom MD2 by FINSEQ_3:34;
then A468: 1 <= len MD2 by FINSEQ_3:27;
A469: ((len MD2) - 1) + (indx D2,D1,(n1 + 1)) = indx D2,D1,j by A446;
for x1 being set st x1 in rng MD2 holds
x1 in (rng MD1) \/ {(D . (i + 1))}

proof

let x1 be set ; :: thesis:
assume A470: x1 in rng MD2 ; :: thesis:
then reconsider x1 = x1 as Real ;
MD2 . 1 = D2 . (indx D2,D1,(n1 + 1)) by A270, A277, A262, A263, A359, FINSEQ_6:124;
then A471: D1 . (n1 + 1) <= x1 by A271, A470, Th15;
MD2 . (len MD2) = D2 . (indx D2,D1,j) by A272, A270, A263, A359, A468, A445, A469, FINSEQ_6:128;
then A472: x1 <= D1 . j by A238, A470, Th15;
A473: rng MD2 c= rng D2 by A359, FINSEQ_6:125;

now

per cases by A14, A470, A473, XBOOLE_0:def 3;

suppose x1 in rng D1 ; :: thesis:

then consider k being Element of NAT such that
A474: k in dom D1 and
A475: D1 . k = x1 by PARTFUN1:26;
A476: n1 + 1 <= k by A268, A471, A474, A475, SEQM_3:def 1;
then A477: 1 <= k - n1 by XREAL_1:21;
n1 <= n1 + 1 by NAT_1:11;
then consider n being Nat such that
A478: k = n1 + n by A476, NAT_1:10, XXREAL_0:2;
A479: k <= j by A236, A472, A474, A475, SEQM_3:def 1;
then k - n1 <= len MD1 by A364, XREAL_1:11;
then n in dom MD1 by A477, A478, FINSEQ_3:27;
then A480: MD1 . n in rng MD1 by FUNCT_1:def 5;
(j - (n1 + 1)) + 1 = j - n1 ;
then A481: k - n1 <= (j - (n1 + 1)) + 1 by A479, XREAL_1:11;
n in NAT by ORDINAL1:def 13;
then MD1 . n = D1 . (((k - n1) - 1) + (n1 + 1)) by A257, A266, A244, A358, A477, A481, A478, FINSEQ_6:128
.= D1 . k ;
hence x1 in (rng MD1) \/ {(D . (i + 1))} by A475, A480, XBOOLE_0:def 3; :: thesis:

end;

suppose x1 in rng D ; :: thesis:

then consider n being Element of NAT such that
A482: n in dom D and
A483: D . n = x1 by PARTFUN1:26;
A484: not i + 1 < n

proof

A485: upper_bound (divset D1,j) = D1 . j

proof

per cases ;

suppose j = 1 ; :: thesis:

hence upper_bound (divset D1,j) = D1 . j by A236, INTEGRA1:def 5; :: thesis:

end;

suppose j <> 1 ; :: thesis:

hence upper_bound (divset D1,j) = D1 . j by A236, INTEGRA1:def 5; :: thesis:

end;

consider y1 being Real such that
A486: y1 = D . (i + 1) ;
A487: D . n in rng D by A482, FUNCT_1:def 5;
assume i + 1 < n ; :: thesis:
then A488: D . (i + 1) < D . n by A235, A482, 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 A488, XXREAL_0:2;
then D . n in divset D1,j by A472, A483, A485, INTEGRA2:1;
then A489: x1 in (rng D) /\ (divset D1,j) by A483, A487, XBOOLE_0:def 4;
D . (i + 1) in rng D by A235, FUNCT_1:def 5;
then y1 in (rng D) /\ (divset D1,j) by A237, A486, XBOOLE_0:def 4;
hence contradiction by A11, A236, A483, A488, A489, A486, Th4; :: thesis:

end;

A490: upper_bound (divset D1,n1) = D1 . n1

proof

per cases ;

suppose n1 = 1 ; :: thesis:

hence upper_bound (divset D1,n1) = D1 . n1 by A241, INTEGRA1:def 5; :: thesis:

end;

suppose n1 <> 1 ; :: thesis:

hence upper_bound (divset D1,n1) = D1 . n1 by A241, INTEGRA1:def 5; :: thesis:

end;

D . i <= upper_bound (divset D1,n1) by A242, INTEGRA2:1;
then D . i < D1 . (n1 + 1) by A354, A490, XXREAL_0:2;
then D . i < D . n by A471, A483, XXREAL_0:2;
then i < n by A240, A482, SEQ_4:154;
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 A483, A484, TARSKI:def 1;
hence x1 in (rng MD1) \/ {(D . (i + 1))} by XBOOLE_0:def 3; :: thesis:

end;

hence x1 in (rng MD1) \/ {(D . (i + 1))} ; :: thesis:

end;

then rng MD2 c= (rng MD1) \/ {(D . (i + 1))} by TARSKI:def 3;
then A491: rng MD2 = (rng MD1) \/ {(D . (i + 1))} by A467, XBOOLE_0:def 10;
delta MD1 in rng (upper_volume (chi B,B),MD1) by XXREAL_2:def 8;
then consider k being Element of NAT such that
A492: k in dom (upper_volume (chi B,B),MD1) and
A493: (upper_volume (chi B,B),MD1) . k = delta MD1 by PARTFUN1:26;
A494: k in Seg (len (upper_volume (chi B,B),MD1)) by A492, FINSEQ_1:def 3;
then A495: k in Seg (len MD1) by INTEGRA1:def 7;
then A496: k in dom MD1 by FINSEQ_1:def 3;
A497: k <= len MD1 by A495, FINSEQ_1:3;
then k + n1 <= j by A364, XREAL_1:21;
then A498: k + n1 <= len D1 by A266, XXREAL_0:2;
A499: 1 <= k by A494, FINSEQ_1:3;
A500: n1 + 1 > 1 by A280, NAT_1:13;
then n1 > 1 - 1 by XREAL_1:21;
then A501: k < k + n1 by XREAL_1:31;
then 1 < k + n1 by A499, XXREAL_0:2;
then A502: k + n1 in dom D1 by A498, FINSEQ_3:27;
( lower_bound (divset MD1,k) = lower_bound (divset D1,(k + n1)) & upper_bound (divset MD1,k) = upper_bound (divset D1,(k + n1)) )

proof

per cases ;

suppose A503: k = 1 ; :: thesis:

then upper_bound (divset MD1,k) = MD1 . k by A496, INTEGRA1:def 5;
then A504: upper_bound (divset MD1,k) = D1 . ((k + (n1 + 1)) - 1) by A257, A266, A244, A358, A361, A499, A497, FINSEQ_6:128;
lower_bound (divset D1,(k + n1)) = D1 . ((k + n1) - 1) by A499, A501, A502, INTEGRA1:def 5;
hence ( lower_bound (divset MD1,k) = lower_bound (divset D1,(k + n1)) & upper_bound (divset MD1,k) = upper_bound (divset D1,(k + n1)) ) by A355, A500, A496, A502, A503, A504, INTEGRA1:def 5; :: thesis:

end;

suppose A505: k <> 1 ; :: thesis:

then upper_bound (divset MD1,k) = MD1 . k by A496, INTEGRA1:def 5;
then A506: upper_bound (divset MD1,k) = D1 . ((k + (n1 + 1)) - 1) by A257, A266, A244, A358, A361, A499, A497, FINSEQ_6:128;
A507: lower_bound (divset MD1,k) = MD1 . (k - 1) by A496, A505, INTEGRA1:def 5;
A508: k - 1 in dom MD1 by A496, A505, INTEGRA1:9;
then A509: k - 1 <= len MD1 by FINSEQ_3:27;
1 <= k - 1 by A508, FINSEQ_3:27;
then lower_bound (divset MD1,k) = D1 . (((k - 1) + (n1 + 1)) - 1) by A257, A266, A244, A358, A361, A508, A509, A507, FINSEQ_6:128;
hence ( lower_bound (divset MD1,k) = lower_bound (divset D1,(k + n1)) & upper_bound (divset MD1,k) = upper_bound (divset D1,(k + n1)) ) by A499, A501, A502, A506, INTEGRA1:def 5; :: thesis:

end;

then divset MD1,k = [.(lower_bound (divset D1,(k + n1))),(upper_bound (divset D1,(k + n1))).] by INTEGRA1:5;
then A510: divset MD1,k = divset D1,(k + n1) by INTEGRA1:5;
k + n1 in Seg (len D1) by A502, FINSEQ_1:def 3;
then k + n1 in Seg (len (upper_volume (chi A,A),D1)) by INTEGRA1:def 7;
then A511: k + n1 in dom (upper_volume (chi A,A),D1) by FINSEQ_1:def 3;
k in dom MD1 by A495, FINSEQ_1:def 3;
then delta MD1 = vol (divset MD1,k) by A493, INTEGRA1:22;
then delta MD1 = (upper_volume (chi A,A),D1) . (k + n1) by A502, A510, INTEGRA1:22;
then delta MD1 in rng (upper_volume (chi A,A),D1) by A511, FUNCT_1:def 5;
then delta MD1 <= max (rng (upper_volume (chi A,A),D1)) by XXREAL_2:def 8;
then A512: delta MD1 <= delta D1 ;
(upper_bound (rng f)) - (lower_bound (rng f)) >= 0 by A1, Lm3, XREAL_1:50;
then A513: ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta MD1) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) by A512, XREAL_1:66;
lower_bound (divset D1,j) <= D . (i + 1) by A237, INTEGRA2:1;
then A514: D1 . (j - 1) <= D . (i + 1) by A236, A245, A280, INTEGRA1:def 5;
A515: D . (i + 1) <= upper_bound (divset D1,j) by A237, INTEGRA2:1;
A516: (indx D2,D1,n1) + 1 <= indx D2,D1,j by A272, A386, XXREAL_0:2;
A517: 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:
assume that
A518: 1 <= k and
A519: k <= len (lower_volume g,MD2) ; :: thesis:
A520: k in Seg (len (lower_volume g,MD2)) by A518, A519, FINSEQ_1:3;
then A521: (mid H₁(D2),((indx D2,D1,n1) + 1),(indx D2,D1,j)) . k = H₁(D2) . ((k + ((indx D2,D1,n1) + 1)) - 1) by A389, A447, A444, A516, A518, A519, FINSEQ_6:128;
1 <= (indx D2,D1,n1) + 1 by NAT_1:12;
then 1 + 1 <= k + ((indx D2,D1,n1) + 1) by A518, XREAL_1:9;
then A522: 1 <= (k + ((indx D2,D1,n1) + 1)) - 1 by XREAL_1:21;
consider k2 being Element of NAT such that
A523: (indx D2,D1,n1) + 1 = 1 + k2 ;
k <= (indx D2,D1,j) - (((indx D2,D1,n1) + 1) - 1) by A446, A391, A519, INTEGRA1:def 8;
then k + (((indx D2,D1,n1) + 1) - 1) <= indx D2,D1,j by XREAL_1:21;
then (k + ((indx D2,D1,n1) + 1)) - 1 <= len H₁(D2) by A444, XXREAL_0:2;
then k + k2 in Seg (len H₁(D2)) by A522, A523, FINSEQ_1:3;
then A524: k + k2 in Seg (len D2) by INTEGRA1:def 8;
then k + k2 in dom D2 by FINSEQ_1:def 3;
then A525: (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 A521, A523, INTEGRA1:def 8;
A526: k in Seg (len MD2) by A520, INTEGRA1:def 8;
A527: ( 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, A518, A523, XREAL_1:9;
then A528: k + k2 > 1 by NAT_1:13;
A529: k in dom MD2 by A526, FINSEQ_1:def 3;
A530: k + k2 in dom D2 by A524, FINSEQ_1:def 3;

per cases ;

suppose A531: k = 1 ; :: thesis:

then upper_bound (divset MD2,k) = MD2 . k by A529, INTEGRA1:def 5;
then A532: upper_bound (divset MD2,k) = D2 . ((k + ((indx D2,D1,n1) + 1)) - 1) by A272, A263, A359, A389, A391, A447, A518, A519, A520, FINSEQ_6:128;
A533: lower_bound (divset D2,(k + k2)) = D2 . ((k + k2) - 1) by A528, A530, INTEGRA1:def 5;
lower_bound (divset MD2,k) = D1 . n1 by A355, A529, A531, INTEGRA1:def 5;
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, A523, A528, A530, A531, A532, A533, INTEGRA1:def 5, INTEGRA1:def 21; :: thesis:

end;

suppose A534: k <> 1 ; :: thesis:

then upper_bound (divset MD2,k) = MD2 . k by A529, INTEGRA1:def 5;
then A535: upper_bound (divset MD2,k) = D2 . ((k + ((indx D2,D1,n1) + 1)) - 1) by A272, A263, A359, A389, A391, A447, A518, A519, A520, FINSEQ_6:128;
A536: k - 1 <= ((indx D2,D1,j) - ((indx D2,D1,n1) + 1)) + 1 by A447, A519, XREAL_1:148, XXREAL_0:2;
A537: lower_bound (divset MD2,k) = MD2 . (k - 1) by A529, A534, INTEGRA1:def 5;
A538: k - 1 in dom MD2 by A529, A534, INTEGRA1:9;
then 1 <= k - 1 by FINSEQ_3:27;
then lower_bound (divset MD2,k) = D2 . (((k - 1) + ((indx D2,D1,n1) + 1)) - 1) by A272, A263, A359, A389, A391, A538, A536, A537, FINSEQ_6:128;
hence ( lower_bound (divset MD2,k) = lower_bound (divset D2,(k + k2)) & upper_bound (divset MD2,k) = upper_bound (divset D2,(k + k2)) ) by A523, A528, A530, A535, INTEGRA1:def 5; :: thesis:

end;

divset MD2,k = [.(lower_bound (divset MD2,k)),(upper_bound (divset MD2,k)).] by INTEGRA1:5;
then A539: divset MD2,k = divset D2,(k + k2) by A527, INTEGRA1:5;
k in dom MD2 by A526, FINSEQ_1:def 3;
then divset D2,(k + k2) c= B by A539, INTEGRA1:10;
then A540: rng (f | (divset D2,(k + k2))) = rng (g | (divset D2,(k + k2))) by FUNCT_1:82;
k in dom MD2 by A526, FINSEQ_1:def 3;
hence (lower_volume g,MD2) . k = (mid H₁(D2),((indx D2,D1,n1) + 1),(indx D2,D1,j)) . k by A525, A539, A540, INTEGRA1:def 8; :: thesis:

end;

lower_bound (divset D1,j) <= D . (i + 1) by A237, INTEGRA2:1;
then A541: D . (i + 1) in divset MD1,(len MD1) by A375, A515, INTEGRA2:1;
j - 1 in dom D1 by A236, A245, A280, INTEGRA1:9;
then D1 . n1 < D1 . (j - 1) by A241, A443, SEQM_3:def 1;
then D . (i + 1) > lower_bound B by A355, A514, 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 A357, A433, A491, A541, A388, Th12;
then A542: (Sum (lower_volume g,MD2)) - (Sum (lower_volume g,MD1)) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta MD1) by A442, XXREAL_0:2;
(indx D2,D1,n1) + 1 <= len H₁(D2) by A387, INTEGRA1:def 8;
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, A389, A444, A516, FINSEQ_6:124;
then len (lower_volume g,MD2) = len (mid H₁(D2),((indx D2,D1,n1) + 1),(indx D2,D1,j)) by A272, A386, A447, XREAL_1:235, XXREAL_0:2;
then A543: Sum (lower_volume g,MD2) = Sum (mid H₁(D2),((indx D2,D1,n1) + 1),(indx D2,D1,j)) by A517, FINSEQ_1:18;
n1 + 1 <= len H₁(D1) by A267, INTEGRA1:def 8;
then len (mid H₁(D1),(n1 + 1),j) = (j -' (n1 + 1)) + 1 by A257, A265, A244, A408, FINSEQ_6:124
.= (j - (n1 + 1)) + 1 by A257, XREAL_1:235 ;
then Sum (lower_volume g,MD1) = Sum (mid H₁(D1),(n1 + 1),j) by A372, A409, FINSEQ_1:18;
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 A542, A513, A543, XXREAL_0:2; :: thesis:

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:

end;

then A544: (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_1:9;
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_4:154;
then A545: 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 8;
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 22
.= Sum (mid H₁(D2),1,(indx D2,D1,n1)) by A260, FINSEQ_6:122 ;
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_1:105
.= Sum (mid H₁(D2),1,(indx D2,D1,j)) by A260, A545, A264, INTEGRA2:4
.= Sum (H₁(D2) | (indx D2,D1,j)) by A262, FINSEQ_6:122 ;
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 22;
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, A544; :: thesis:

end;

hence S₁[i + 1] ; :: thesis:

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:

end;

A546: len D1 in dom D1 by FINSEQ_5:6;
then D1 . (len D1) = D2 . (indx D2,D1,(len D1)) by A13, INTEGRA1:def 21;
then upper_bound A = D2 . (indx D2,D1,(len D1)) by INTEGRA1:def 2;
then A547: 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
A548: j in dom D1 and
A549: D . (len D) in divset D1,j and
A550: H₂(D2, indx D2,D1,j) - H₂(D1,j) <= ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by A16;
A551: j = len D1

proof

assume A552: j <> len D1 ; :: thesis:
j <= len D1 by A548, FINSEQ_3:27;
then j < len D1 by A552, XXREAL_0:1;
then D1 . j < D1 . (len D1) by A548, A546, SEQM_3:def 1;
then A553: D1 . j < upper_bound A by INTEGRA1:def 2;
A554: upper_bound (divset D1,j) < upper_bound A

proof

per cases ;

suppose j = 1 ; :: thesis:

hence upper_bound (divset D1,j) < upper_bound A by A548, A553, INTEGRA1:def 5; :: thesis:

end;

suppose j <> 1 ; :: thesis:

hence upper_bound (divset D1,j) < upper_bound A by A548, A553, INTEGRA1:def 5; :: thesis:

end;

D . (len D) <= upper_bound (divset D1,j) by A549, INTEGRA2:1;
hence contradiction by A554, INTEGRA1:def 2; :: thesis:

end;

indx D2,D1,(len D1) in dom D2 by A13, A546, INTEGRA1:def 21;
then indx D2,D1,(len D1) = len D2 by A15, A547, SEQ_4:155;
then H₂(D2, len D2) - (lower_sum f,D1) <= ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by A550, A551, INTEGRA1:45;
hence (lower_sum f,D2) - (lower_sum f,D1) <= ((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (delta D1) by INTEGRA1:45; :: thesis:

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:

end;

A555: lim (delta T) = 0 by A2, FDIFF_1:def 1;
A556: delta T is non-empty by A2, FDIFF_1:def 1;
A557: 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:
assume e > 0 ; :: thesis:
then consider n being Element of NAT such that
A558: for m being Element of NAT st n <= m holds
abs (((delta T) . m) - 0 ) < e by A4, A555, 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:
A559: (delta T) . m = delta (T . m) by DelDel;
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
A560: i in dom (upper_volume (chi A,A),(T . m)) and
A561: delta (T . m) = (upper_volume (chi A,A),(T . m)) . i by PARTFUN1:26;
consider D being Division of A such that
A562: D = T . m ;
i in Seg (len (upper_volume (chi A,A),(T . m))) by A560, FINSEQ_1:def 3;
then i in Seg (len D) by A562, INTEGRA1:def 7;
then i in dom D by FINSEQ_1:def 3;
then A563: delta (T . m) = vol (divset (T . m),i) by A561, A562, INTEGRA1:22;
assume n <= m ; :: thesis:
then abs (((delta T) . m) - 0 ) < e by A558;
then A564: ((delta T) . m) + (abs (((delta T) . m) - 0 )) < e + (abs (((delta T) . m) - 0 )) by ABSVALUE:11, XREAL_1:10;
(delta T) . m <> 0 by A556, SEQ_1:7;
hence ( 0 < (delta T) . m & (delta T) . m < e ) by A564, A559, A563, INTEGRA1:11, XREAL_1:8; :: thesis:

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:

end;

A565: 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:
assume A566: e > 0 ; :: thesis:
then A567: e / 2 > 0 by XREAL_1:141;
reconsider e = e as Real by XREAL_0:def 1;
A568: (upper_bound (rng f)) - (lower_bound (rng f)) >= 0 by A1, Lm3, XREAL_1:50;
not dom (lower_sum_set f) is empty by INTEGRA1:def 12;
then A569: not rng (lower_sum_set f) is empty by RELAT_1:65;
A570: 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 16;
then consider y being real number such that
A571: y in rng (lower_sum_set f) and
A572: (lower_integral f) - (e / 2) < y by A567, A570, A569, SEQ_4:def 4;
consider D being Division of A such that
A573: D in dom (lower_sum_set f) and
A574: y = (lower_sum_set f) . D and
A575: D . 1 > lower_bound A by A3, A571, Lm7;
deffunc H₁( Nat) -> Element of REAL = vol (divset D,$1);
set p = len D;
consider v being FinSequence of REAL such that
A576: ( 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
A577: v,v1 are_fiberwise_equipotent by INTEGRA2:3;
defpred S₁[ Nat] means ( $1 in dom v1 & v1 . $1 > 0 );
A578: dom v = Seg (len D) by A576, FINSEQ_1:def 3;
A579: 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
A580: v = v1 * H by A577, CLASSES1:85;
consider k being Element of NAT such that
A581: k in dom D and
A582: vol (divset D,k) > 0 by A3, Th1;
A583: dom D = Seg (len v) by A576, FINSEQ_1:def 3;
then H . k in dom v1 by A576, A578, A580, A581, FUNCT_1:21;
then reconsider Hk = H . k as Nat ;
v . k > 0 by A576, A578, A581, A582, A583;
then S₁[Hk] by A576, A578, A580, A581, A583, FUNCT_1:21, FUNCT_1:22;
hence ex k being Nat st S₁[k] ; :: thesis:

end;

consider k being Nat such that
A584: ( S₁[k] & ( for n being Nat st S₁[n] holds
k <= n ) ) from NAT_1:sch 5(A579);
A585: 2 * (len D) > 0 by XREAL_1:131;
then A586: (2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1) > 0 by A568, XREAL_1:131;
min (v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1))) > 0

proof

per cases by XXREAL_0:15;

suppose min (v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1))) = v1 . k ; :: thesis:

hence min (v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1))) > 0 by A584; :: thesis:

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:

hence min (v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1))) > 0 by A566, A586, XREAL_1:141; :: thesis:

end;

then consider n being Element of NAT such that
A587: 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 A557;
take n ; :: thesis:
A588: y = lower_sum f,D by A573, A574, INTEGRA1:def 12;
for m being Element of NAT st n <= m holds
abs (((lower_sum f,T) . m) - (lower_integral f)) < e

proof

A589: v1 . 1 > 0

proof

A590: 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:
assume A591: n1 in dom D ; :: thesis:
then A592: 1 <= n1 by FINSEQ_3:27;

per cases by A592, XXREAL_0:1;

suppose A593: n1 = 1 ; :: thesis:

then A594: upper_bound (divset D,n1) = D . n1 by A591, INTEGRA1:def 5;
lower_bound (divset D,n1) = lower_bound A by A591, A593, INTEGRA1:def 5;
then vol (divset D,n1) = (D . n1) - (lower_bound A) by A594, INTEGRA1:def 6;
hence vol (divset D,n1) > 0 by A575, A593, XREAL_1:52; :: thesis:

end;

suppose A595: n1 > 1 ; :: thesis:

then A596: upper_bound (divset D,n1) = D . n1 by A591, INTEGRA1:def 5;
lower_bound (divset D,n1) = D . (n1 - 1) by A591, A595, INTEGRA1:def 5;
then A597: vol (divset D,n1) = (D . n1) - (D . (n1 - 1)) by A596, INTEGRA1:def 6;
n1 < n1 + 1 by XREAL_1:31;
then A598: n1 - 1 < n1 by XREAL_1:21;
n1 - 1 in dom D by A591, A595, INTEGRA1:9;
then D . (n1 - 1) < D . n1 by A591, A598, SEQM_3:def 1;
hence vol (divset D,n1) > 0 by A597, XREAL_1:52; :: thesis:

end;

A599: k <= len v1 by A584, FINSEQ_3:27;
1 <= k by A584, FINSEQ_3:27;
then 1 <= len v1 by A599, XXREAL_0:2;
then 1 in dom v1 by FINSEQ_3:27;
then A600: v1 . 1 in rng v1 by FUNCT_1:def 5;
rng v = rng v1 by A577, CLASSES1:83;
then consider n1 being Element of NAT such that
A601: n1 in dom v and
A602: v1 . 1 = v . n1 by A600, PARTFUN1:26;
n1 in Seg (len D) by A576, A601, FINSEQ_1:def 3;
then A603: n1 in dom D by FINSEQ_1:def 3;
v1 . 1 = vol (divset D,n1) by A576, A601, A602;
hence v1 . 1 > 0 by A590, A603; :: thesis:

end;

A604: v1 . k = min (rng (upper_volume (chi A,A),D))

proof

A605: k = 1

proof

len v1 = len v by A577, RFINSEQ:16;
then k in Seg (len v) by A584, FINSEQ_1:def 3;
then A606: 1 <= k by FINSEQ_1:3;
k in Seg (len v1) by A584, FINSEQ_1:def 3;
then k <= len v1 by FINSEQ_1:3;
then 1 <= len v1 by A606, XXREAL_0:2;
then A607: 1 in dom v1 by FINSEQ_3:27;
assume k <> 1 ; :: thesis:
then k > 1 by A606, XXREAL_0:1;
hence contradiction by A584, A589, A607; :: thesis:

end;

end;

end;

hence for m being Element of NAT st n <= m holds
abs (((lower_sum f,T) . m) - (lower_integral f)) < e ; :: thesis:

end;

hence lower_sum f,T is convergent by SEQ_2:def 6; :: thesis:
hence lim (lower_sum f,T) = lower_integral f by A565, SEQ_2:def 7; :: thesis: