:: INTEGRA3 semantic presentation
Lemma1:
for b1 being Real
for b2 being FinSequence of REAL holds
( b1 in dom b2 implies ( 1 <= b1 & b1 <= len b2 ) )
Lemma2:
for b1 being Real holds b1 - 1 <= b1
by XREAL_1:148;
Lemma3:
for b1 being Nat holds (b1 -' b1) + 1 = 1
Lemma4:
for b1 being Nat holds
not ( 1 <= b1 & b1 <= 2 & not b1 = 1 & not b1 = 2 )
theorem Th1: :: INTEGRA3:1
theorem Th2: :: INTEGRA3:2
theorem Th3: :: INTEGRA3:3
theorem Th4: :: INTEGRA3:4
theorem Th5: :: INTEGRA3:5
theorem Th6: :: INTEGRA3:6
theorem Th7: :: INTEGRA3:7
theorem Th8: :: INTEGRA3:8
Lemma13:
for b1 being closed-interval Subset of REAL
for b2 being Function of b1, REAL holds
( b2 is_bounded_on b1 implies sup (rng b2) >= inf (rng b2) )
Lemma14:
for b1, b2 being closed-interval Subset of REAL
for b3 being Function of b1, REAL holds
( b3 is_bounded_on b1 & b2 c= b1 implies ( inf (rng (b3 | b2)) >= inf (rng b3) & inf (rng b3) <= sup (rng (b3 | b2)) & sup (rng (b3 | b2)) <= sup (rng b3) & inf (rng (b3 | b2)) <= sup (rng b3) ) )
Lemma15:
for b1 being Nat
for b2 being closed-interval Subset of REAL
for b3 being Element of divs b2 holds
( b1 in dom b3 implies vol (divset b3,b1) <= delta b3 )
Lemma16:
for b1 being Real
for b2 being closed-interval Subset of REAL
for b3 being Nat
for b4, b5 being Element of divs b2 holds
( b3 = (len b4) - 1 & b1 in divset b4,(len b4) & len b4 >= 2 & b4 <= b5 & rng b5 = (rng b4) \/ {b1} implies rng (b5 | (indx b5,b4,b3)) = rng (b4 | b3) )
theorem Th9: :: INTEGRA3:9
theorem Th10: :: INTEGRA3:10
Lemma19:
for b1 being Real
for b2 being closed-interval Subset of REAL
for b3 being PartFunc of b2, REAL holds
not ( vol b2 <> 0 & b1 in rng (lower_sum_set b3) & ( for b4 being Element of divs b2 holds
not ( b4 in dom (lower_sum_set b3) & b1 = (lower_sum_set b3) . b4 & b4 . 1 > inf b2 ) ) )
theorem Th11: :: INTEGRA3:11
for b
1 being
closed-interval Subset of
REAL for b
2 being
Element of
divs b
1for b
3 being
Realfor b
4, b
5 being
Nat holds
not ( b
4 in dom b
2 & b
5 in dom b
2 & b
4 <= b
5 & b
3 < (mid b2,b4,b5) . 1 & ( for b
6 being
closed-interval Subset of
REAL holds
not ( b
3 = inf b
6 &
sup b
6 = (mid b2,b4,b5) . (len (mid b2,b4,b5)) &
len (mid b2,b4,b5) = (b5 - b4) + 1 &
mid b
2,b
4,b
5 is
DivisionPoint of b
6 ) ) )
Lemma21:
for b1 being closed-interval Subset of REAL
for b2 being Element of divs b1 holds
( vol b1 <> 0 & len b2 = 1 implies <*(inf b1)*> ^ b2 is non empty increasing FinSequence of REAL )
Lemma22:
for b1 being closed-interval Subset of REAL
for b2 being Element of divs b1 holds
( inf b1 < b2 . 1 implies <*(inf b1)*> ^ b2 is non empty increasing FinSequence of REAL )
Lemma23:
for b1 being closed-interval Subset of REAL
for b2 being PartFunc of b1, REAL
for b3, b4 being Element of divs b1 holds
( b4 = <*(inf b1)*> ^ b3 implies ( ( for b5 being Nat holds
( b5 in Seg (len b3) implies divset b4,(b5 + 1) = divset b3,b5 ) ) & upper_volume b2,b3 = (upper_volume b2,b4) /^ 1 & lower_volume b2,b3 = (lower_volume b2,b4) /^ 1 ) )
Lemma24:
for b1 being closed-interval Subset of REAL
for b2, b3 being Element of divs b1 holds
( b3 = <*(inf b1)*> ^ b2 implies vol (divset b3,1) = 0 )
Lemma25:
for b1 being closed-interval Subset of REAL
for b2 being Function of b1, REAL
for b3, b4 being Element of divs b1 holds
( b4 = <*(inf b1)*> ^ b3 implies delta b4 = delta b3 )
theorem Th12: :: INTEGRA3:12
theorem Th13: :: INTEGRA3:13
theorem Th14: :: INTEGRA3:14
for b
1 being
closed-interval Subset of
REAL for b
2, b
3 being
Element of
divs b
1for b
4 being
Realfor b
5, b
6 being
Nat holds
not ( b
5 in dom b
2 & b
6 in dom b
2 & b
5 <= b
6 & b
2 <= b
3 & b
4 < (mid b3,(indx b3,b2,b5),(indx b3,b2,b6)) . 1 & ( for b
7 being
closed-interval Subset of
REAL for b
8, b
9 being
Element of
divs b
7 holds
not ( b
4 = inf b
7 &
sup b
7 = b
9 . (len b9) &
sup b
7 = b
8 . (len b8) & b
8 <= b
9 & b
8 = mid b
2,b
5,b
6 & b
9 = mid b
3,
(indx b3,b2,b5),
(indx b3,b2,b6) ) ) )
theorem Th15: :: INTEGRA3:15
theorem Th16: :: INTEGRA3:16
theorem Th17: :: INTEGRA3:17
theorem Th18: :: INTEGRA3:18
theorem Th19: :: INTEGRA3:19
theorem Th20: :: INTEGRA3:20