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;