Article INTEGRA6, MML version 4.99.1005
:: INTEGRA6:th 1
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4 & b1 + b3 = b2 + b4
holds b1 = b2 & b3 = b4;
:: INTEGRA6:th 2
theorem
for b1, b2, b3 being real set
st b1 <= b2
holds ].b3 - b1,b3 + b1.[ c= ].b3 - b2,b3 + b2.[;
:: INTEGRA6:th 3
theorem
for b1 being Relation-like set
for b2, b3, b4 being set
st b2 c= b3 & b2 c= b4
holds (b1 | b3) | b2 = (b1 | b4) | b2;
:: INTEGRA6:th 4
theorem
for b1, b2, b3 being set
st b1 c= b2 & b1 c= b3
holds (chi(b2,b2)) | b1 = (chi(b3,b3)) | b1;
:: INTEGRA6:th 5
theorem
for b1, b2 being real set
st b1 <= b2
holds vol ['b1,b2'] = b2 - b1;
:: INTEGRA6:th 6
theorem
for b1, b2 being real set holds
vol ['min(b1,b2),max(b1,b2)'] = abs (b2 - b1);
:: INTEGRA6:th 7
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
st b1 c= proj1 b2 & b2 is_integrable_on b1 & b2 is_bounded_on b1
holds abs b2 is_integrable_on b1 &
abs integral(b2,b1) <= integral(abs b2,b1);
:: INTEGRA6:th 8
theorem
for b1, b2 being real set
for b3 being Function-like Relation of REAL,REAL
st b1 <= b2 & ['b1,b2'] c= proj1 b3 & b3 is_integrable_on ['b1,b2'] & b3 is_bounded_on ['b1,b2']
holds abs integral(b3,b1,b2) <= integral(abs b3,b1,b2);
:: INTEGRA6:th 9
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
st b1 c= proj1 b2 & b2 is_integrable_on b1 & b2 is_bounded_on b1
holds b3 (#) b2 is_integrable_on b1 &
integral(b3 (#) b2,b1) = b3 * integral(b2,b1);
:: INTEGRA6:th 10
theorem
for b1, b2, b3 being real set
for b4 being Function-like Relation of REAL,REAL
st b1 <= b2 & ['b1,b2'] c= proj1 b4 & b4 is_integrable_on ['b1,b2'] & b4 is_bounded_on ['b1,b2']
holds integral(b3 (#) b4,b1,b2) = b3 * integral(b4,b1,b2);
:: INTEGRA6:th 11
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b1 c= proj1 b2 & b1 c= proj1 b3 & b2 is_integrable_on b1 & b2 is_bounded_on b1 & b3 is_integrable_on b1 & b3 is_bounded_on b1
holds b2 + b3 is_integrable_on b1 &
b2 - b3 is_integrable_on b1 &
integral(b2 + b3,b1) = (integral(b2,b1)) + integral(b3,b1) &
integral(b2 - b3,b1) = (integral(b2,b1)) - integral(b3,b1);
:: INTEGRA6:th 12
theorem
for b1, b2 being real set
for b3, b4 being Function-like Relation of REAL,REAL
st b1 <= b2 & ['b1,b2'] c= proj1 b3 & ['b1,b2'] c= proj1 b4 & b3 is_integrable_on ['b1,b2'] & b4 is_integrable_on ['b1,b2'] & b3 is_bounded_on ['b1,b2'] & b4 is_bounded_on ['b1,b2']
holds integral(b3 + b4,b1,b2) = (integral(b3,b1,b2)) + integral(b4,b1,b2) &
integral(b3 - b4,b1,b2) = (integral(b3,b1,b2)) - integral(b4,b1,b2);
:: INTEGRA6:th 13
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_bounded_on b1 & b3 is_bounded_on b1
holds b2 (#) b3 is_bounded_on b1;
:: INTEGRA6:th 14
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b1 c= proj1 b2 & b1 c= proj1 b3 & b2 is_integrable_on b1 & b2 is_bounded_on b1 & b3 is_integrable_on b1 & b3 is_bounded_on b1
holds b2 (#) b3 is_integrable_on b1;
:: INTEGRA6:th 15
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of NAT
st 0 < b2 & 0 < vol b1
holds ex b3 being Element of divs b1 st
len b3 = b2 &
(for b4 being Element of NAT
st b4 in dom b3
holds b3 . b4 = (inf b1) + (((vol b1) / b2) * b4));
:: INTEGRA6:th 16
theorem
for b1 being closed-interval Element of bool REAL
st 0 < vol b1
holds ex b2 being Function-like quasi_total Relation of NAT,divs b1 st
delta b2 is convergent &
lim delta b2 = 0 &
(for b3 being Element of NAT holds
ex b4 being Element of divs b1 st
b4 divide_into_equal b3 + 1 & b2 . b3 = b4);
:: INTEGRA6:th 17
theorem
for b1, b2, b3 being real set
for b4 being Function-like Relation of REAL,REAL
st b1 <= b2 & b4 is_integrable_on ['b1,b2'] & b4 is_bounded_on ['b1,b2'] & ['b1,b2'] c= proj1 b4 & b3 in ['b1,b2']
holds b4 is_integrable_on ['b1,b3'] &
b4 is_integrable_on ['b3,b2'] &
integral(b4,b1,b2) = (integral(b4,b1,b3)) + integral(b4,b3,b2);
:: INTEGRA6:th 18
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like Relation of REAL,REAL
st b1 <= b2 & b2 <= b3 & b3 <= b4 & b5 is_integrable_on ['b1,b4'] & b5 is_bounded_on ['b1,b4'] & ['b1,b4'] c= proj1 b5
holds b5 is_integrable_on ['b2,b3'] & b5 is_bounded_on ['b2,b3'] & ['b2,b3'] c= proj1 b5;
:: INTEGRA6:th 19
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Function-like Relation of REAL,REAL
st b1 <= b2 & b2 <= b3 & b3 <= b4 & b5 is_integrable_on ['b1,b4'] & b6 is_integrable_on ['b1,b4'] & b5 is_bounded_on ['b1,b4'] & b6 is_bounded_on ['b1,b4'] & ['b1,b4'] c= proj1 b5 & ['b1,b4'] c= proj1 b6
holds b5 + b6 is_integrable_on ['b2,b3'] & b5 + b6 is_bounded_on ['b2,b3'];
:: INTEGRA6:th 20
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like Relation of REAL,REAL
st b1 <= b2 & b5 is_integrable_on ['b1,b2'] & b5 is_bounded_on ['b1,b2'] & ['b1,b2'] c= proj1 b5 & b3 in ['b1,b2'] & b4 in ['b1,b2']
holds integral(b5,b1,b4) = (integral(b5,b1,b3)) + integral(b5,b3,b4);
:: INTEGRA6:th 21
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like Relation of REAL,REAL
st b1 <= b2 & b5 is_integrable_on ['b1,b2'] & b5 is_bounded_on ['b1,b2'] & ['b1,b2'] c= proj1 b5 & b3 in ['b1,b2'] & b4 in ['b1,b2']
holds ['min(b3,b4),max(b3,b4)'] c= proj1 abs b5 &
abs b5 is_integrable_on ['min(b3,b4),max(b3,b4)'] &
abs b5 is_bounded_on ['min(b3,b4),max(b3,b4)'] &
abs integral(b5,b3,b4) <= integral(abs b5,min(b3,b4),max(b3,b4));
:: INTEGRA6:th 22
theorem
for b1, b2, b3, b4 being real set
for b5 being Function-like Relation of REAL,REAL
st b1 <= b2 & b3 <= b4 & b5 is_integrable_on ['b1,b2'] & b5 is_bounded_on ['b1,b2'] & ['b1,b2'] c= proj1 b5 & b3 in ['b1,b2'] & b4 in ['b1,b2']
holds ['b3,b4'] c= proj1 abs b5 &
abs b5 is_integrable_on ['b3,b4'] &
abs b5 is_bounded_on ['b3,b4'] &
abs integral(b5,b3,b4) <= integral(abs b5,b3,b4) &
abs integral(b5,b4,b3) <= integral(abs b5,b3,b4);
:: INTEGRA6:th 23
theorem
for b1, b2, b3, b4, b5 being real set
for b6 being Function-like Relation of REAL,REAL
st b1 <= b2 &
b3 <= b4 &
b6 is_integrable_on ['b1,b2'] &
b6 is_bounded_on ['b1,b2'] &
['b1,b2'] c= proj1 b6 &
b3 in ['b1,b2'] &
b4 in ['b1,b2'] &
(for b7 being real set
st b7 in ['b3,b4']
holds abs (b6 . b7) <= b5)
holds abs integral(b6,b3,b4) <= b5 * (b4 - b3) &
abs integral(b6,b4,b3) <= b5 * (b4 - b3);
:: INTEGRA6:th 24
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Function-like Relation of REAL,REAL
st b1 <= b2 & b5 is_integrable_on ['b1,b2'] & b6 is_integrable_on ['b1,b2'] & b5 is_bounded_on ['b1,b2'] & b6 is_bounded_on ['b1,b2'] & ['b1,b2'] c= proj1 b5 & ['b1,b2'] c= proj1 b6 & b3 in ['b1,b2'] & b4 in ['b1,b2']
holds integral(b5 + b6,b3,b4) = (integral(b5,b3,b4)) + integral(b6,b3,b4) &
integral(b5 - b6,b3,b4) = (integral(b5,b3,b4)) - integral(b6,b3,b4);
:: INTEGRA6:th 25
theorem
for b1, b2, b3, b4, b5 being real set
for b6 being Function-like Relation of REAL,REAL
st b1 <= b2 & b6 is_integrable_on ['b1,b2'] & b6 is_bounded_on ['b1,b2'] & ['b1,b2'] c= proj1 b6 & b3 in ['b1,b2'] & b4 in ['b1,b2']
holds integral(b5 (#) b6,b3,b4) = b5 * integral(b6,b3,b4);
:: INTEGRA6:th 26
theorem
for b1, b2, b3 being real set
for b4 being Function-like Relation of REAL,REAL
st b1 <= b2 &
['b1,b2'] c= proj1 b4 &
(for b5 being real set
st b5 in ['b1,b2']
holds b4 . b5 = b3)
holds b4 is_integrable_on ['b1,b2'] & b4 is_bounded_on ['b1,b2'] & integral(b4,b1,b2) = b3 * (b2 - b1);
:: INTEGRA6:th 27
theorem
for b1, b2, b3, b4, b5 being real set
for b6 being Function-like Relation of REAL,REAL
st b1 <= b2 &
(for b7 being real set
st b7 in ['b1,b2']
holds b6 . b7 = b3) &
['b1,b2'] c= proj1 b6 &
b4 in ['b1,b2'] &
b5 in ['b1,b2']
holds integral(b6,b4,b5) = b3 * (b5 - b4);
:: INTEGRA6:th 28
theorem
for b1, b2 being real set
for b3 being Function-like Relation of REAL,REAL
for b4 being real set
for b5 being Function-like Relation of REAL,REAL
st b1 <= b2 &
b3 is_integrable_on ['b1,b2'] &
b3 is_bounded_on ['b1,b2'] &
['b1,b2'] c= proj1 b3 &
].b1,b2.[ c= proj1 b5 &
(for b6 being real set
st b6 in ].b1,b2.[
holds b5 . b6 = integral(b3,b1,b6)) &
b4 in ].b1,b2.[ &
b3 is_continuous_in b4
holds b5 is_differentiable_in b4 & diff(b5,b4) = b3 . b4;
:: INTEGRA6:th 29
theorem
for b1, b2 being real set
for b3 being Function-like Relation of REAL,REAL
for b4 being real set
st b1 <= b2 & b3 is_integrable_on ['b1,b2'] & b3 is_bounded_on ['b1,b2'] & ['b1,b2'] c= proj1 b3 & b4 in ].b1,b2.[ & b3 is_continuous_in b4
holds ex b5 being Function-like Relation of REAL,REAL st
].b1,b2.[ c= proj1 b5 &
(for b6 being real set
st b6 in ].b1,b2.[
holds b5 . b6 = integral(b3,b1,b6)) &
b5 is_differentiable_in b4 &
diff(b5,b4) = b3 . b4;