Article INTEGRA4, MML version 4.99.1005

:: INTEGRA4:th 1
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of divs b1
      st vol b1 = 0
   holds len b2 = 1;

:: INTEGRA4:th 2
theorem
for b1 being closed-interval Element of bool REAL holds
   chi(b1,b1) is_integrable_on b1 & integral chi(b1,b1) = vol b1;

:: INTEGRA4:th 3
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL holds
      b2 is total(b1, REAL) & rng b2 = {b3}
   iff
      b2 = b3 (#) chi(b1,b1);

:: INTEGRA4:th 4
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being Element of REAL
      st rng b2 = {b3}
   holds b2 is_integrable_on b1 & integral b2 = b3 * vol b1;

:: INTEGRA4:th 5
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of REAL holds
   ex b3 being Function-like quasi_total Relation of b1,REAL st
      rng b3 = {b2} & b3 is_bounded_on b1;

:: INTEGRA4:th 6
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like Relation of b1,REAL
for b3 being Element of divs b1
      st vol b1 = 0
   holds b2 is_integrable_on b1 & integral b2 = 0;

:: INTEGRA4:th 7
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 & b2 is_integrable_on b1
   holds ex b3 being Element of REAL st
      inf rng b2 <= b3 & b3 <= sup rng b2 & integral b2 = b3 * vol b1;

:: INTEGRA4:th 8
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being Function-like quasi_total Relation of NAT,divs b1
      st b2 is_bounded_on b1 & delta b3 is convergent & lim delta b3 = 0
   holds lower_sum(b2,b3) is convergent & lim lower_sum(b2,b3) = lower_integral b2;

:: INTEGRA4:th 9
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Function-like quasi_total Relation of b1,REAL
for b3 being Function-like quasi_total Relation of NAT,divs b1
      st b2 is_bounded_on b1 & delta b3 is convergent & lim delta b3 = 0
   holds upper_sum(b2,b3) is convergent & lim upper_sum(b2,b3) = upper_integral b2;

:: INTEGRA4:th 10
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 b2 is_upper_integrable_on b1 & b2 is_lower_integrable_on b1;

:: INTEGRA4:prednot 1 => INTEGRA4:pred 1
definition
  let a1 be closed-interval Element of bool REAL;
  let a2 be Element of divs a1;
  let a3 be Element of NAT;
  pred A2 divide_into_equal A3 means
    len a2 = a3 &
     (for b1 being Element of NAT
           st b1 in dom a2
        holds a2 . b1 = (inf a1) + (((vol a1) / len a2) * b1));
end;

:: INTEGRA4:dfs 1
definiens
  let a1 be closed-interval Element of bool REAL;
  let a2 be Element of divs a1;
  let a3 be Element of NAT;
To prove
     a2 divide_into_equal a3
it is sufficient to prove
  thus len a2 = a3 &
     (for b1 being Element of NAT
           st b1 in dom a2
        holds a2 . b1 = (inf a1) + (((vol a1) / len a2) * b1));

:: INTEGRA4:def 1
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of divs b1
for b3 being Element of NAT holds
      b2 divide_into_equal b3
   iff
      len b2 = b3 &
       (for b4 being Element of NAT
             st b4 in dom b2
          holds b2 . b4 = (inf b1) + (((vol b1) / len b2) * b4));

:: INTEGRA4:th 11
theorem
for b1 being closed-interval Element of bool REAL holds
   ex b2 being Function-like quasi_total Relation of NAT,divs b1 st
      delta b2 is convergent & lim delta b2 = 0;

:: INTEGRA4:th 12
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    b2 is_integrable_on b1
   iff
      for b3 being Function-like quasi_total Relation of NAT,divs b1
            st delta b3 is convergent & lim delta b3 = 0
         holds (lim upper_sum(b2,b3)) - lim lower_sum(b2,b3) = 0;

:: INTEGRA4:th 13
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,REAL holds
   max+ b2 is total(b1, REAL) & max- b2 is total(b1, REAL);

:: INTEGRA4:th 14
theorem
for b1 being non empty set
for b2 being set
for b3 being Function-like Relation of b1,REAL
      st b3 is_bounded_above_on b2
   holds max+ b3 is_bounded_above_on b2;

:: INTEGRA4:th 15
theorem
for b1 being non empty set
for b2 being set
for b3 being Function-like Relation of b1,REAL holds
   max+ b3 is_bounded_below_on b2;

:: INTEGRA4:th 16
theorem
for b1 being non empty set
for b2 being set
for b3 being Function-like Relation of b1,REAL
      st b3 is_bounded_below_on b2
   holds max- b3 is_bounded_above_on b2;

:: INTEGRA4:th 17
theorem
for b1 being non empty set
for b2 being set
for b3 being Function-like Relation of b1,REAL holds
   max- b3 is_bounded_below_on b2;

:: INTEGRA4:th 18
theorem
for b1 being closed-interval Element of bool REAL
for b2 being set
for b3 being Function-like Relation of b1,REAL
      st b3 is_bounded_above_on b1
   holds rng (b3 | b2) is bounded_above;

:: INTEGRA4:th 19
theorem
for b1 being closed-interval Element of bool REAL
for b2 being set
for b3 being Function-like Relation of b1,REAL
      st b3 is_bounded_below_on b1
   holds rng (b3 | b2) is bounded_below;

:: INTEGRA4:th 20
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 & b2 is_integrable_on b1
   holds max+ b2 is_integrable_on b1;

:: INTEGRA4:th 21
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
   max- b2 = max+ - b2;

:: INTEGRA4:th 22
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 & b2 is_integrable_on b1
   holds max- b2 is_integrable_on b1;

:: INTEGRA4:th 23
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 & b2 is_integrable_on b1
   holds abs b2 is_integrable_on b1 & abs integral b2 <= integral abs b2;

:: INTEGRA4:th 24
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
      st for b4, b5 being Element of REAL
              st b4 in b2 & b5 in b2
           holds abs ((b3 . b4) - (b3 . b5)) <= b1
   holds (sup rng b3) - inf rng b3 <= b1;

:: INTEGRA4:th 25
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3, b4 being Function-like quasi_total Relation of b2,REAL
      st b3 is_bounded_on b2 &
         0 <= b1 &
         (for b5, b6 being Element of REAL
               st b5 in b2 & b6 in b2
            holds abs ((b4 . b5) - (b4 . b6)) <= b1 * abs ((b3 . b5) - (b3 . b6)))
   holds (sup rng b4) - inf rng b4 <= b1 * ((sup rng b3) - inf rng b3);

:: INTEGRA4:th 26
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3, b4, b5 being Function-like quasi_total Relation of b2,REAL
      st b3 is_bounded_on b2 &
         b4 is_bounded_on b2 &
         0 <= b1 &
         (for b6, b7 being Element of REAL
               st b6 in b2 & b7 in b2
            holds abs ((b5 . b6) - (b5 . b7)) <= b1 * ((abs ((b3 . b6) - (b3 . b7))) + abs ((b4 . b6) - (b4 . b7))))
   holds (sup rng b5) - inf rng b5 <= b1 * (((sup rng b3) - inf rng b3) + ((sup rng b4) - inf rng b4));

:: INTEGRA4:th 27
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3, b4 being Function-like quasi_total Relation of b2,REAL
      st b3 is_bounded_on b2 &
         b3 is_integrable_on b2 &
         b4 is_bounded_on b2 &
         0 < b1 &
         (for b5, b6 being Element of REAL
               st b5 in b2 & b6 in b2
            holds abs ((b4 . b5) - (b4 . b6)) <= b1 * abs ((b3 . b5) - (b3 . b6)))
   holds b4 is_integrable_on b2;

:: INTEGRA4:th 28
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3, b4, b5 being Function-like quasi_total Relation of b2,REAL
      st b3 is_bounded_on b2 &
         b3 is_integrable_on b2 &
         b4 is_bounded_on b2 &
         b4 is_integrable_on b2 &
         b5 is_bounded_on b2 &
         0 < b1 &
         (for b6, b7 being Element of REAL
               st b6 in b2 & b7 in b2
            holds abs ((b5 . b6) - (b5 . b7)) <= b1 * ((abs ((b3 . b6) - (b3 . b7))) + abs ((b4 . b6) - (b4 . b7))))
   holds b5 is_integrable_on b2;

:: INTEGRA4:th 29
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 & b2 is_integrable_on b1 & b3 is_bounded_on b1 & b3 is_integrable_on b1
   holds b2 (#) b3 is_integrable_on b1;

:: INTEGRA4:th 30
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 & b2 is_integrable_on b1 & not 0 in rng b2 & b2 ^ is_bounded_on b1
   holds b2 ^ is_integrable_on b1;