Article INTEGRA5, MML version 4.99.1005
:: INTEGRA5:th 1
theorem
for b1, b2, b3 being FinSequence of REAL
for b4, b5 being Element of REAL
st (b2 = <*b4*> ^ b1 or b2 = b1 ^ <*b4*>) &
(b3 = <*b5*> ^ b1 or b3 = b1 ^ <*b5*>)
holds Sum (b2 - b3) = b4 - b5;
:: INTEGRA5:th 2
theorem
for b1, b2 being FinSequence of REAL
st len b1 = len b2
holds len (b1 + b2) = len b1 &
len (b1 - b2) = len b1 &
Sum (b1 + b2) = (Sum b1) + Sum b2 &
Sum (b1 - b2) = (Sum b1) - Sum b2;
:: INTEGRA5:th 3
theorem
for b1, b2 being FinSequence of REAL
st len b1 = len b2 &
(for b3 being Element of NAT
st b3 in dom b1
holds b1 . b3 <= b2 . b3)
holds Sum b1 <= Sum b2;
:: INTEGRA5:funcnot 1 => INTEGRA5:func 1
definition
let a1 be non empty Element of bool REAL;
let a2 be Function-like Relation of REAL,REAL;
func A2 || A1 -> Function-like Relation of a1,REAL equals
a2 | a1;
end;
:: INTEGRA5:def 1
theorem
for b1 being non empty Element of bool REAL
for b2 being Function-like Relation of REAL,REAL holds
b2 || b1 = b2 | b1;
:: INTEGRA5:th 4
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being non empty Element of bool REAL holds
(b1 || b3) (#) (b2 || b3) = (b1 (#) b2) || b3;
:: INTEGRA5:th 5
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being non empty Element of bool REAL holds
(b1 + b2) || b3 = (b1 || b3) + (b2 || b3);
:: INTEGRA5:prednot 1 => INTEGRA5:pred 1
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of REAL,REAL;
pred A2 is_integrable_on A1 means
a2 || a1 is_integrable_on a1;
end;
:: INTEGRA5:dfs 2
definiens
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of REAL,REAL;
To prove
a2 is_integrable_on a1
it is sufficient to prove
thus a2 || a1 is_integrable_on a1;
:: INTEGRA5:def 2
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL holds
b2 is_integrable_on b1
iff
b2 || b1 is_integrable_on b1;
:: INTEGRA5:funcnot 2 => INTEGRA5:func 2
definition
let a1 be closed-interval Element of bool REAL;
let a2 be Function-like Relation of REAL,REAL;
func integral(A2,A1) -> Element of REAL equals
integral (a2 || a1);
end;
:: INTEGRA5:def 3
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL holds
integral(b2,b1) = integral (b2 || b1);
:: INTEGRA5:th 6
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= dom b2
holds b2 || b1 is total(b1, REAL);
:: INTEGRA5:th 7
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_bounded_above_on b1
holds b2 || b1 is_bounded_above_on b1;
:: INTEGRA5:th 8
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_bounded_below_on b1
holds b2 || b1 is_bounded_below_on b1;
:: INTEGRA5:th 9
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_bounded_on b1
holds b2 || b1 is_bounded_on b1;
:: INTEGRA5:th 10
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_continuous_on b1
holds b2 is_bounded_on b1;
:: INTEGRA5:th 11
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_continuous_on b1
holds b2 is_integrable_on b1;
:: INTEGRA5:th 12
theorem
for b1 being closed-interval Element of bool REAL
for b2 being set
for b3 being Function-like Relation of REAL,REAL
for b4 being Element of divs b1
st b1 c= b2 & b3 is_differentiable_on b2 & b3 `| b2 is_bounded_on b1
holds lower_sum((b3 `| b2) || b1,b4) <= (b3 . sup b1) - (b3 . inf b1) &
(b3 . sup b1) - (b3 . inf b1) <= upper_sum((b3 `| b2) || b1,b4);
:: INTEGRA5:th 13
theorem
for b1 being closed-interval Element of bool REAL
for b2 being set
for b3 being Function-like Relation of REAL,REAL
st b1 c= b2 & b3 is_differentiable_on b2 & b3 `| b2 is_integrable_on b1 & b3 `| b2 is_bounded_on b1
holds integral(b3 `| b2,b1) = (b3 . sup b1) - (b3 . inf b1);
:: INTEGRA5:th 14
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_non_decreasing_on b1 & b1 c= dom b2
holds rng (b2 | b1) is bounded;
:: INTEGRA5:th 15
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_non_decreasing_on b1 & b1 c= dom b2
holds inf rng (b2 | b1) = b2 . inf b1 &
sup rng (b2 | b1) = b2 . sup b1;
:: INTEGRA5:th 16
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_monotone_on b1 & b1 c= dom b2
holds b2 is_integrable_on b1;
:: INTEGRA5:th 17
theorem
for b1 being Function-like Relation of REAL,REAL
for b2, b3 being closed-interval Element of bool REAL
st b1 is_continuous_on b2 & b3 c= b2
holds b1 is_integrable_on b3;
:: INTEGRA5:th 18
theorem
for b1 being Function-like Relation of REAL,REAL
for b2, b3, b4 being closed-interval Element of bool REAL
for b5 being set
st b2 c= b5 & b1 is_differentiable_on b5 & b1 `| b5 is_continuous_on b2 & inf b2 = inf b3 & sup b3 = inf b4 & sup b4 = sup b2
holds b3 c= b2 &
b4 c= b2 &
integral(b1 `| b5,b2) = (integral(b1 `| b5,b3)) + integral(b1 `| b5,b4);
:: INTEGRA5:funcnot 3 => INTEGRA5:func 3
definition
let a1, a2 be real set;
assume a1 <= a2;
func ['A1,A2'] -> closed-interval Element of bool REAL equals
[.a1,a2.];
end;
:: INTEGRA5:def 4
theorem
for b1, b2 being real set
st b1 <= b2
holds ['b1,b2'] = [.b1,b2.];
:: INTEGRA5:funcnot 4 => INTEGRA5:func 4
definition
let a1, a2 be real set;
let a3 be Function-like Relation of REAL,REAL;
func integral(A3,A1,A2) -> Element of REAL equals
integral(a3,['a1,a2'])
if a1 <= a2
otherwise - integral(a3,['a2,a1']);
end;
:: INTEGRA5:def 5
theorem
for b1, b2 being real set
for b3 being Function-like Relation of REAL,REAL holds
(b1 <= b2 implies integral(b3,b1,b2) = integral(b3,['b1,b2'])) &
(b1 <= b2 or integral(b3,b1,b2) = - integral(b3,['b2,b1']));
:: INTEGRA5:th 19
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being closed-interval Element of bool REAL
for b3, b4 being Element of REAL
st b2 = [.b3,b4.]
holds integral(b1,b2) = integral(b1,b3,b4);
:: INTEGRA5:th 20
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being closed-interval Element of bool REAL
for b3, b4 being Element of REAL
st b2 = [.b4,b3.]
holds - integral(b1,b2) = integral(b1,b3,b4);
:: INTEGRA5:th 21
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
for b4 being open Element of bool REAL
st b2 is_differentiable_on b4 & b3 is_differentiable_on b4 & b1 c= b4 & b2 `| b4 is_integrable_on b1 & b2 `| b4 is_bounded_on b1 & b3 `| b4 is_integrable_on b1 & b3 `| b4 is_bounded_on b1
holds integral((b2 `| b4) (#) b3,b1) = (((b2 . sup b1) * (b3 . sup b1)) - ((b2 . inf b1) * (b3 . inf b1))) - integral(b2 (#) (b3 `| b4),b1);