Article INTEGRA1, MML version 4.99.1005
:: INTEGRA1:attrnot 1 => INTEGRA1:attr 1
definition
let a1 be Element of bool REAL;
attr a1 is closed-interval means
ex b1, b2 being Element of REAL st
b1 <= b2 & a1 = [.b1,b2.];
end;
:: INTEGRA1:dfs 1
definiens
let a1 be Element of bool REAL;
To prove
a1 is closed-interval
it is sufficient to prove
thus ex b1, b2 being Element of REAL st
b1 <= b2 & a1 = [.b1,b2.];
:: INTEGRA1:def 1
theorem
for b1 being Element of bool REAL holds
b1 is closed-interval
iff
ex b2, b3 being Element of REAL st
b2 <= b3 & b1 = [.b2,b3.];
:: INTEGRA1:exreg 1
registration
cluster complex-membered ext-real-membered real-membered closed-interval Element of bool REAL;
end;
:: INTEGRA1:th 1
theorem
for b1 being closed-interval Element of bool REAL holds
b1 is compact;
:: INTEGRA1:th 2
theorem
for b1 being closed-interval Element of bool REAL holds
b1 is not empty;
:: INTEGRA1:condreg 1
registration
cluster closed-interval -> non empty compact (Element of bool REAL);
end;
:: INTEGRA1:th 3
theorem
for b1 being closed-interval Element of bool REAL holds
b1 is bounded_below & b1 is bounded_above;
:: INTEGRA1:condreg 2
registration
cluster closed-interval -> bounded (Element of bool REAL);
end;
:: INTEGRA1:exreg 2
registration
cluster complex-membered ext-real-membered real-membered closed-interval Element of bool REAL;
end;
:: INTEGRA1:th 4
theorem
for b1 being closed-interval Element of bool REAL holds
ex b2, b3 being Element of REAL st
b2 <= b3 & b2 = inf b1 & b3 = sup b1;
:: INTEGRA1:th 5
theorem
for b1 being closed-interval Element of bool REAL holds
b1 = [.inf b1,sup b1.];
:: INTEGRA1:th 6
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3, b4, b5 being real set
st b1 = [.b2,b4.] & b1 = [.b3,b5.]
holds b2 = b3 & b4 = b5;
:: INTEGRA1:modenot 1 => INTEGRA1:mode 1
definition
let a1 be non empty compact Element of bool REAL;
mode DivisionPoint of A1 -> non empty increasing FinSequence of REAL means
rng it c= a1 & it . len it = sup a1;
end;
:: INTEGRA1:dfs 2
definiens
let a1 be non empty compact Element of bool REAL;
let a2 be non empty increasing FinSequence of REAL;
To prove
a2 is DivisionPoint of a1
it is sufficient to prove
thus rng a2 c= a1 & a2 . len a2 = sup a1;
:: INTEGRA1:def 2
theorem
for b1 being non empty compact Element of bool REAL
for b2 being non empty increasing FinSequence of REAL holds
b2 is DivisionPoint of b1
iff
rng b2 c= b1 & b2 . len b2 = sup b1;
:: INTEGRA1:funcnot 1 => INTEGRA1:func 1
definition
let a1 be non empty compact Element of bool REAL;
func divs A1 -> set means
for b1 being set holds
b1 in it
iff
b1 is DivisionPoint of a1;
end;
:: INTEGRA1:def 3
theorem
for b1 being non empty compact Element of bool REAL
for b2 being set holds
b2 = divs b1
iff
for b3 being set holds
b3 in b2
iff
b3 is DivisionPoint of b1;
:: INTEGRA1:funcreg 1
registration
let a1 be non empty compact Element of bool REAL;
cluster divs a1 -> non empty;
end;
:: INTEGRA1:modenot 2 => INTEGRA1:mode 2
definition
let a1 be non empty compact Element of bool REAL;
mode Division of A1 -> non empty set means
for b1 being set holds
b1 in it
iff
b1 is DivisionPoint of a1;
end;
:: INTEGRA1:dfs 4
definiens
let a1 be non empty compact Element of bool REAL;
let a2 be non empty set;
To prove
a2 is Division of a1
it is sufficient to prove
thus for b1 being set holds
b1 in a2
iff
b1 is DivisionPoint of a1;
:: INTEGRA1:def 4
theorem
for b1 being non empty compact Element of bool REAL
for b2 being non empty set holds
b2 is Division of b1
iff
for b3 being set holds
b3 in b2
iff
b3 is DivisionPoint of b1;
:: INTEGRA1:exreg 3
registration
let a1 be non empty compact Element of bool REAL;
cluster non empty Division of a1;
end;
:: INTEGRA1:modenot 3 => INTEGRA1:mode 3
definition
let a1 be non empty compact Element of bool REAL;
let a2 be non empty Division of a1;
redefine mode Element of a2 -> DivisionPoint of a1;
end;
:: INTEGRA1:th 8
theorem
for b1 being Element of NAT
for b2 being closed-interval Element of bool REAL
for b3 being non empty Division of b2
for b4 being Element of b3
st b1 in dom b4
holds b4 . b1 in b2;
:: INTEGRA1:th 9
theorem
for b1 being Element of NAT
for b2 being closed-interval Element of bool REAL
for b3 being non empty Division of b2
for b4 being Element of b3
st b1 in dom b4 & b1 <> 1
holds b1 - 1 in dom b4 & b4 . (b1 - 1) in b2 & b1 - 1 in NAT;
:: INTEGRA1:funcnot 2 => INTEGRA1:func 2
definition
let a1 be closed-interval Element of bool REAL;
let a2 be non empty Division of a1;
let a3 be Element of a2;
let a4 be natural set;
assume a4 in dom a3;
func divset(A3,A4) -> closed-interval Element of bool REAL means
inf it = inf a1 & sup it = a3 . a4
if a4 = 1
otherwise inf it = a3 . (a4 - 1) & sup it = a3 . a4;
end;
:: INTEGRA1:def 5
theorem
for b1 being closed-interval Element of bool REAL
for b2 being non empty Division of b1
for b3 being Element of b2
for b4 being natural set
st b4 in dom b3
for b5 being closed-interval Element of bool REAL holds
(b4 = 1 implies (b5 = divset(b3,b4)
iff
inf b5 = inf b1 & sup b5 = b3 . b4)) &
(b4 = 1 or (b5 = divset(b3,b4)
iff
inf b5 = b3 . (b4 - 1) & sup b5 = b3 . b4));
:: INTEGRA1:th 10
theorem
for b1 being Element of NAT
for b2 being closed-interval Element of bool REAL
for b3 being non empty Division of b2
for b4 being Element of b3
st b1 in dom b4
holds divset(b4,b1) c= b2;
:: INTEGRA1:funcnot 3 => INTEGRA1:func 3
definition
let a1 be Element of bool REAL;
func vol A1 -> Element of REAL equals
(sup a1) - inf a1;
end;
:: INTEGRA1:def 6
theorem
for b1 being Element of bool REAL holds
vol b1 = (sup b1) - inf b1;
:: INTEGRA1:th 11
theorem
for b1 being non empty bounded Element of bool REAL holds
0 <= vol b1;
:: INTEGRA1:funcnot 4 => INTEGRA1:func 4
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
let a3 be non empty Division of a1;
let a4 be Element of a3;
func upper_volume(A2,A4) -> FinSequence of REAL means
len it = len a4 &
(for b1 being natural set
st b1 in dom a4
holds it . b1 = (sup rng (a2 | divset(a4,b1))) * vol divset(a4,b1));
end;
:: INTEGRA1:def 7
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being non empty Division of b1
for b4 being Element of b3
for b5 being FinSequence of REAL holds
b5 = upper_volume(b2,b4)
iff
len b5 = len b4 &
(for b6 being natural set
st b6 in dom b4
holds b5 . b6 = (sup rng (b2 | divset(b4,b6))) * vol divset(b4,b6));
:: INTEGRA1:funcnot 5 => INTEGRA1:func 5
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
let a3 be non empty Division of a1;
let a4 be Element of a3;
func lower_volume(A2,A4) -> FinSequence of REAL means
len it = len a4 &
(for b1 being natural set
st b1 in dom a4
holds it . b1 = (inf rng (a2 | divset(a4,b1))) * vol divset(a4,b1));
end;
:: INTEGRA1:def 8
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being non empty Division of b1
for b4 being Element of b3
for b5 being FinSequence of REAL holds
b5 = lower_volume(b2,b4)
iff
len b5 = len b4 &
(for b6 being natural set
st b6 in dom b4
holds b5 . b6 = (inf rng (b2 | divset(b4,b6))) * vol divset(b4,b6));
:: INTEGRA1:funcnot 6 => INTEGRA1:func 6
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
let a3 be non empty Division of a1;
let a4 be Element of a3;
func upper_sum(A2,A4) -> Element of REAL equals
Sum upper_volume(a2,a4);
end;
:: INTEGRA1:def 9
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being non empty Division of b1
for b4 being Element of b3 holds
upper_sum(b2,b4) = Sum upper_volume(b2,b4);
:: INTEGRA1:funcnot 7 => INTEGRA1:func 7
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
let a3 be non empty Division of a1;
let a4 be Element of a3;
func lower_sum(A2,A4) -> Element of REAL equals
Sum lower_volume(a2,a4);
end;
:: INTEGRA1:def 10
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being non empty Division of b1
for b4 being Element of b3 holds
lower_sum(b2,b4) = Sum lower_volume(b2,b4);
:: INTEGRA1:funcnot 8 => INTEGRA1:func 8
definition
let a1 be closed-interval Element of bool REAL;
redefine func divs a1 -> Division of a1;
end;
:: INTEGRA1:funcnot 9 => INTEGRA1:func 9
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
func upper_sum_set A2 -> Function-like Relation of divs a1,REAL means
dom it = divs a1 &
(for b1 being Element of divs a1
st b1 in dom it
holds it . b1 = upper_sum(a2,b1));
end;
:: INTEGRA1:def 11
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being Function-like Relation of divs b1,REAL holds
b3 = upper_sum_set b2
iff
dom b3 = divs b1 &
(for b4 being Element of divs b1
st b4 in dom b3
holds b3 . b4 = upper_sum(b2,b4));
:: INTEGRA1:funcnot 10 => INTEGRA1:func 10
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
func lower_sum_set A2 -> Function-like Relation of divs a1,REAL means
dom it = divs a1 &
(for b1 being Element of divs a1
st b1 in dom it
holds it . b1 = lower_sum(a2,b1));
end;
:: INTEGRA1:def 12
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being Function-like Relation of divs b1,REAL holds
b3 = lower_sum_set b2
iff
dom b3 = divs b1 &
(for b4 being Element of divs b1
st b4 in dom b3
holds b3 . b4 = lower_sum(b2,b4));
:: INTEGRA1:prednot 1 => INTEGRA1:pred 1
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
pred A2 is_upper_integrable_on A1 means
rng upper_sum_set a2 is bounded_below;
end;
:: INTEGRA1:dfs 13
definiens
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
To prove
a2 is_upper_integrable_on a1
it is sufficient to prove
thus rng upper_sum_set a2 is bounded_below;
:: INTEGRA1:def 13
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL holds
b2 is_upper_integrable_on b1
iff
rng upper_sum_set b2 is bounded_below;
:: INTEGRA1:prednot 2 => INTEGRA1:pred 2
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
pred A2 is_lower_integrable_on A1 means
rng lower_sum_set a2 is bounded_above;
end;
:: INTEGRA1:dfs 14
definiens
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
To prove
a2 is_lower_integrable_on a1
it is sufficient to prove
thus rng lower_sum_set a2 is bounded_above;
:: INTEGRA1:def 14
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL holds
b2 is_lower_integrable_on b1
iff
rng lower_sum_set b2 is bounded_above;
:: INTEGRA1:funcnot 11 => INTEGRA1:func 11
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
func upper_integral A2 -> Element of REAL equals
inf rng upper_sum_set a2;
end;
:: INTEGRA1:def 15
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL holds
upper_integral b2 = inf rng upper_sum_set b2;
:: INTEGRA1:funcnot 12 => INTEGRA1:func 12
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
func lower_integral A2 -> Element of REAL equals
sup rng lower_sum_set a2;
end;
:: INTEGRA1:def 16
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL holds
lower_integral b2 = sup rng lower_sum_set b2;
:: INTEGRA1:prednot 3 => INTEGRA1:pred 3
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
pred A2 is_integrable_on A1 means
a2 is_upper_integrable_on a1 & a2 is_lower_integrable_on a1 & upper_integral a2 = lower_integral a2;
end;
:: INTEGRA1:dfs 17
definiens
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
To prove
a2 is_integrable_on a1
it is sufficient to prove
thus a2 is_upper_integrable_on a1 & a2 is_lower_integrable_on a1 & upper_integral a2 = lower_integral a2;
:: INTEGRA1:def 17
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL holds
b2 is_integrable_on b1
iff
b2 is_upper_integrable_on b1 & b2 is_lower_integrable_on b1 & upper_integral b2 = lower_integral b2;
:: INTEGRA1:funcnot 13 => INTEGRA1:func 13
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
func integral A2 -> Element of REAL equals
upper_integral a2;
end;
:: INTEGRA1:def 18
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL holds
integral b2 = upper_integral b2;
:: INTEGRA1:th 12
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL holds
rng (b2 + b3) c= (rng b2) + rng b3;
:: INTEGRA1:th 13
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
st b2 is_bounded_below_on b1
holds rng b2 is bounded_below;
:: INTEGRA1:th 14
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
st rng b2 is bounded_below
holds b2 is_bounded_below_on b1;
:: INTEGRA1:th 15
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
st b2 is_bounded_above_on b1
holds rng b2 is bounded_above;
:: INTEGRA1:th 16
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
st rng b2 is bounded_above
holds b2 is_bounded_above_on b1;
:: INTEGRA1:th 17
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
st b2 is_bounded_on b1
holds rng b2 is bounded;
:: INTEGRA1:th 18
theorem
for b1 being non empty set holds
chi(b1,b1) is_constant_on b1;
:: INTEGRA1:th 19
theorem
for b1 being non empty set
for b2 being non empty Element of bool b1 holds
rng chi(b2,b2) = {1};
:: INTEGRA1:th 20
theorem
for b1 being non empty set
for b2 being non empty Element of bool b1
for b3 being set
st b3 meets dom chi(b2,b2)
holds rng ((chi(b2,b2)) | b3) = {1};
:: INTEGRA1:th 21
theorem
for b1 being Element of NAT
for b2 being closed-interval Element of bool REAL
for b3 being non empty Division of b2
for b4 being Element of b3
st b1 in dom b4
holds vol divset(b4,b1) = (lower_volume(chi(b2,b2),b4)) . b1;
:: INTEGRA1:th 22
theorem
for b1 being Element of NAT
for b2 being closed-interval Element of bool REAL
for b3 being non empty Division of b2
for b4 being Element of b3
st b1 in dom b4
holds vol divset(b4,b1) = (upper_volume(chi(b2,b2),b4)) . b1;
:: INTEGRA1:th 23
theorem
for b1, b2, b3 being FinSequence of REAL
st len b1 = len b2 &
len b1 = len b3 &
(for b4 being Element of NAT
st b4 in dom b1
holds b3 . b4 = (b1 /. b4) + (b2 /. b4))
holds Sum b3 = (Sum b1) + Sum b2;
:: INTEGRA1:th 24
theorem
for b1, b2, b3 being FinSequence of REAL
st len b1 = len b2 &
len b1 = len b3 &
(for b4 being Element of NAT
st b4 in dom b1
holds b3 . b4 = (b1 /. b4) - (b2 /. b4))
holds Sum b3 = (Sum b1) - Sum b2;
:: INTEGRA1:th 25
theorem
for b1 being closed-interval Element of bool REAL
for b2 being non empty Division of b1
for b3 being Element of b2 holds
Sum lower_volume(chi(b1,b1),b3) = vol b1;
:: INTEGRA1:th 26
theorem
for b1 being closed-interval Element of bool REAL
for b2 being non empty Division of b1
for b3 being Element of b2 holds
Sum upper_volume(chi(b1,b1),b3) = vol b1;
:: INTEGRA1:funcnot 14 => INTEGRA1:func 14
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
let a3 be non empty Division of a1;
let a4 be Element of a3;
redefine func upper_volume(a2,a4) -> non empty FinSequence of REAL;
end;
:: INTEGRA1:funcnot 15 => INTEGRA1:func 15
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of a1,REAL;
let a3 be non empty Division of a1;
let a4 be Element of a3;
redefine func lower_volume(a2,a4) -> non empty FinSequence of REAL;
end;
:: INTEGRA1:th 27
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4 being Element of b3
st b2 is_bounded_below_on b1
holds (inf rng b2) * vol b1 <= lower_sum(b2,b4);
:: INTEGRA1:th 28
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4 being Element of b3
for b5 being Element of NAT
st b2 is_bounded_above_on b1 & b5 in dom b4
holds (sup rng (b2 | divset(b4,b5))) * vol divset(b4,b5) <= (sup rng b2) * vol divset(b4,b5);
:: INTEGRA1:th 29
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4 being Element of b3
st b2 is_bounded_above_on b1
holds upper_sum(b2,b4) <= (sup rng b2) * vol b1;
:: INTEGRA1:th 30
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4 being Element of b3
st b2 is_bounded_on b1
holds lower_sum(b2,b4) <= upper_sum(b2,b4);
:: INTEGRA1:funcnot 16 => INTEGRA1:func 16
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Element of divs a1;
func delta A2 -> Element of REAL equals
upper_bound rng upper_volume(chi(a1,a1),a2);
end;
:: INTEGRA1:def 19
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of divs b1 holds
delta b2 = upper_bound rng upper_volume(chi(b1,b1),b2);
:: INTEGRA1:prednot 4 => INTEGRA1:pred 4
definition
let a1 be closed-interval Element of bool REAL;
let a2 be non empty Division of a1;
let a3, a4 be Element of a2;
pred A3 <= A4 means
len a3 <= len a4 & rng a3 c= rng a4;
end;
:: INTEGRA1:dfs 20
definiens
let a1 be closed-interval Element of bool REAL;
let a2 be non empty Division of a1;
let a3, a4 be Element of a2;
To prove
a3 <= a4
it is sufficient to prove
thus len a3 <= len a4 & rng a3 c= rng a4;
:: INTEGRA1:def 20
theorem
for b1 being closed-interval Element of bool REAL
for b2 being non empty Division of b1
for b3, b4 being Element of b2 holds
b3 <= b4
iff
len b3 <= len b4 & rng b3 c= rng b4;
:: INTEGRA1:prednot 5 => INTEGRA1:pred 4
notation
let a1 be closed-interval Element of bool REAL;
let a2 be non empty Division of a1;
let a3, a4 be Element of a2;
synonym a4 >= a3 for a3 <= a4;
end;
:: INTEGRA1:th 31
theorem
for b1 being closed-interval Element of bool REAL
for b2 being non empty Division of b1
for b3, b4 being Element of b2
st len b3 = 1
holds b3 <= b4;
:: INTEGRA1:th 32
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4, b5 being Element of b3
st b2 is_bounded_above_on b1 & len b4 = 1
holds upper_sum(b2,b5) <= upper_sum(b2,b4);
:: INTEGRA1:th 33
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4, b5 being Element of b3
st b2 is_bounded_below_on b1 & len b4 = 1
holds lower_sum(b2,b4) <= lower_sum(b2,b5);
:: INTEGRA1:th 34
theorem
for b1 being Element of NAT
for b2 being closed-interval Element of bool REAL
for b3 being non empty Division of b2
for b4 being Element of b3
st b1 in dom b4
holds ex b5, b6 being closed-interval Element of bool REAL st
b5 = [.inf b2,b4 . b1.] & b6 = [.b4 . b1,sup b2.] & b2 = b5 \/ b6;
:: INTEGRA1:th 35
theorem
for b1 being Element of NAT
for b2 being closed-interval Element of bool REAL
for b3 being non empty Division of b2
for b4, b5 being Element of b3
st b1 in dom b4 & b4 <= b5
holds ex b6 being Element of NAT st
b6 in dom b5 & b4 . b1 = b5 . b6;
:: INTEGRA1:funcnot 17 => INTEGRA1:func 17
definition
let a1 be closed-interval Element of bool REAL;
let a2 be non empty Division of a1;
let a3, a4 be Element of a2;
let a5 be natural set;
assume a3 <= a4;
func indx(A4,A3,A5) -> Element of NAT means
it in dom a4 & a3 . a5 = a4 . it
if a5 in dom a3
otherwise it = 0;
end;
:: INTEGRA1:def 21
theorem
for b1 being closed-interval Element of bool REAL
for b2 being non empty Division of b1
for b3, b4 being Element of b2
for b5 being natural set
st b3 <= b4
for b6 being Element of NAT holds
(b5 in dom b3 implies (b6 = indx(b4,b3,b5)
iff
b6 in dom b4 & b3 . b5 = b4 . b6)) &
(b5 in dom b3 or (b6 = indx(b4,b3,b5)
iff
b6 = 0));
:: INTEGRA1:th 36
theorem
for b1 being increasing FinSequence of REAL
for b2 being Element of NAT
st b2 <= len b1
holds b1 /^ b2 is increasing FinSequence of REAL;
:: INTEGRA1:th 37
theorem
for b1 being increasing FinSequence of REAL
for b2, b3 being Element of NAT
st b3 in dom b1 & b2 <= b3
holds mid(b1,b2,b3) is increasing FinSequence of REAL;
:: INTEGRA1:th 38
theorem
for b1 being closed-interval Element of bool REAL
for b2 being non empty Division of b1
for b3 being Element of b2
for b4, b5 being Element of NAT
st b4 in dom b3 & b5 in dom b3 & b4 <= b5
holds ex b6 being closed-interval Element of bool REAL st
inf b6 = (mid(b3,b4,b5)) . 1 &
sup b6 = (mid(b3,b4,b5)) . len mid(b3,b4,b5) &
len mid(b3,b4,b5) = (b5 - b4) + 1 &
mid(b3,b4,b5) is DivisionPoint of b6;
:: INTEGRA1:th 39
theorem
for b1, b2 being closed-interval Element of bool REAL
for b3 being non empty Division of b1
for b4 being non empty Division of b2
for b5 being Element of b3
for b6, b7 being Element of NAT
st b6 in dom b5 & b7 in dom b5 & b6 <= b7 & inf b2 <= b5 . b6 & b5 . b7 = sup b2
holds mid(b5,b6,b7) is Element of b4;
:: INTEGRA1:funcnot 18 => INTEGRA1:func 18
definition
let a1 be FinSequence of REAL;
func PartSums A1 -> FinSequence of REAL means
len it = len a1 &
(for b1 being natural set
st b1 in dom a1
holds it . b1 = Sum (a1 | b1));
end;
:: INTEGRA1:def 22
theorem
for b1, b2 being FinSequence of REAL holds
b2 = PartSums b1
iff
len b2 = len b1 &
(for b3 being natural set
st b3 in dom b1
holds b2 . b3 = Sum (b1 | b3));
:: INTEGRA1:th 40
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4, b5 being Element of b3
st b4 <= b5 & b2 is_bounded_above_on b1
for b6 being non empty Element of NAT
st b6 in dom b4
holds Sum ((upper_volume(b2,b5)) | indx(b5,b4,b6)) <= Sum ((upper_volume(b2,b4)) | b6);
:: INTEGRA1:th 41
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4, b5 being Element of b3
st b4 <= b5 & b2 is_bounded_below_on b1
for b6 being non empty Element of NAT
st b6 in dom b4
holds Sum ((lower_volume(b2,b4)) | b6) <= Sum ((lower_volume(b2,b5)) | indx(b5,b4,b6));
:: INTEGRA1:th 42
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4, b5 being Element of b3
for b6 being Element of NAT
st b4 <= b5 & b6 in dom b4 & b2 is_bounded_above_on b1
holds (PartSums upper_volume(b2,b5)) . indx(b5,b4,b6) <= (PartSums upper_volume(b2,b4)) . b6;
:: INTEGRA1:th 43
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4, b5 being Element of b3
for b6 being Element of NAT
st b4 <= b5 & b6 in dom b4 & b2 is_bounded_below_on b1
holds (PartSums lower_volume(b2,b4)) . b6 <= (PartSums lower_volume(b2,b5)) . indx(b5,b4,b6);
:: INTEGRA1:th 44
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being non empty Division of b1
for b4 being Element of b3 holds
(PartSums upper_volume(b2,b4)) . len b4 = upper_sum(b2,b4);
:: INTEGRA1:th 45
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being non empty Division of b1
for b4 being Element of b3 holds
(PartSums lower_volume(b2,b4)) . len b4 = lower_sum(b2,b4);
:: INTEGRA1:th 46
theorem
for b1 being closed-interval Element of bool REAL
for b2 being non empty Division of b1
for b3, b4 being Element of b2
st b3 <= b4
holds indx(b4,b3,len b3) = len b4;
:: INTEGRA1:th 47
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4, b5 being Element of b3
st b4 <= b5 & b2 is_bounded_above_on b1
holds upper_sum(b2,b5) <= upper_sum(b2,b4);
:: INTEGRA1:th 48
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4, b5 being Element of b3
st b4 <= b5 & b2 is_bounded_below_on b1
holds lower_sum(b2,b4) <= lower_sum(b2,b5);
:: INTEGRA1:th 49
theorem
for b1 being closed-interval Element of bool REAL
for b2 being non empty Division of b1
for b3, b4 being Element of b2 holds
ex b5 being Element of b2 st
b3 <= b5 & b4 <= b5;
:: INTEGRA1:th 50
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being non empty Division of b1
for b4, b5 being Element of b3
st b2 is_bounded_on b1
holds lower_sum(b2,b4) <= upper_sum(b2,b5);
:: INTEGRA1:th 51
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
st b2 is_bounded_on b1
holds lower_integral b2 <= upper_integral b2;
:: INTEGRA1:th 52
theorem
for b1, b2 being Element of bool REAL holds
(- b1) + - b2 = - (b1 + b2);
:: INTEGRA1:th 53
theorem
for b1, b2 being Element of bool REAL
st b1 is bounded_above & b2 is bounded_above
holds b1 + b2 is bounded_above;
:: INTEGRA1:th 54
theorem
for b1, b2 being non empty Element of bool REAL
st b1 is bounded_above & b2 is bounded_above
holds sup (b1 + b2) = (sup b1) + sup b2;
:: INTEGRA1:th 55
theorem
for b1 being Element of NAT
for b2 being closed-interval Element of bool REAL
for b3, b4 being Function-like quasi_total Relation of b2,REAL
for b5 being non empty Division of b2
for b6 being Element of b5
st b1 in dom b6 & b3 is_bounded_above_on b2 & b4 is_bounded_above_on b2
holds (upper_volume(b3 + b4,b6)) . b1 <= ((upper_volume(b3,b6)) . b1) + ((upper_volume(b4,b6)) . b1);
:: INTEGRA1:th 56
theorem
for b1 being Element of NAT
for b2 being closed-interval Element of bool REAL
for b3, b4 being Function-like quasi_total Relation of b2,REAL
for b5 being non empty Division of b2
for b6 being Element of b5
st b1 in dom b6 & b3 is_bounded_below_on b2 & b4 is_bounded_below_on b2
holds ((lower_volume(b3,b6)) . b1) + ((lower_volume(b4,b6)) . b1) <= (lower_volume(b3 + b4,b6)) . b1;
:: INTEGRA1:th 57
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Function-like quasi_total Relation of b1,REAL
for b4 being non empty Division of b1
for b5 being Element of b4
st b2 is_bounded_above_on b1 & b3 is_bounded_above_on b1
holds upper_sum(b2 + b3,b5) <= (upper_sum(b2,b5)) + upper_sum(b3,b5);
:: INTEGRA1:th 58
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Function-like quasi_total Relation of b1,REAL
for b4 being non empty Division of b1
for b5 being Element of b4
st b2 is_bounded_below_on b1 & b3 is_bounded_below_on b1
holds (lower_sum(b2,b5)) + lower_sum(b3,b5) <= lower_sum(b2 + b3,b5);
:: INTEGRA1:th 59
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Function-like quasi_total Relation of b1,REAL
st b2 is_bounded_on b1 & b3 is_bounded_on b1 & b2 is_integrable_on b1 & b3 is_integrable_on b1
holds b2 + b3 is_integrable_on b1 &
integral (b2 + b3) = (integral b2) + integral b3;