Article INTEGRA3, MML version 4.99.1005

:: INTEGRA3:th 1
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of divs b1
      st vol b1 <> 0
   holds ex b3 being Element of NAT st
      b3 in dom b2 & 0 < vol divset(b2,b3);

:: INTEGRA3:th 2
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Element of divs b2
      st b1 in b2
   holds ex b4 being Element of NAT st
      b4 in dom b3 & b1 in divset(b3,b4);

:: INTEGRA3:th 3
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Element of divs b1 holds
ex b4 being Element of divs b1 st
   b2 <= b4 & b3 <= b4 & rng b4 = (rng b2) \/ rng b3;

:: INTEGRA3:th 4
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Element of divs b1
   st delta b3 < lower_bound rng upper_volume(chi(b1,b1),b2)
for b4, b5 being Element of REAL
for b6 being Element of NAT
      st b6 in dom b3 & b4 in (rng b2) /\ divset(b3,b6) & b5 in (rng b2) /\ divset(b3,b6)
   holds b4 = b5;

:: INTEGRA3:th 5
theorem
for b1, b2 being FinSequence of REAL
      st rng b1 = rng b2 & b1 is increasing & b2 is increasing
   holds b1 = b2;

:: INTEGRA3:th 6
theorem
for b1, b2 being Element of NAT
for b3 being closed-interval Element of bool REAL
for b4, b5 being Element of divs b3
      st b4 <= b5 & b1 in dom b4 & b2 in dom b4 & b1 <= b2
   holds indx(b5,b4,b1) <= indx(b5,b4,b2) & indx(b5,b4,b1) in dom b5;

:: INTEGRA3:th 7
theorem
for b1, b2 being Element of NAT
for b3 being closed-interval Element of bool REAL
for b4, b5 being Element of divs b3
      st b4 <= b5 & b1 in dom b4 & b2 in dom b4 & b1 < b2
   holds indx(b5,b4,b1) < indx(b5,b4,b2);

:: INTEGRA3:th 8
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of divs b1 holds
   0 <= delta b2;

:: INTEGRA3:th 9
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
for b4, b5 being Element of divs b2
      st b1 in divset(b4,len b4) &
         2 <= len b4 &
         b4 <= b5 &
         rng b5 = (rng b4) \/ {b1} &
         b3 is_bounded_on b2
   holds (Sum lower_volume(b3,b5)) - Sum lower_volume(b3,b4) <= ((sup rng b3) - inf rng b3) * delta b4;

:: INTEGRA3:th 10
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
for b4, b5 being Element of divs b2
      st b1 in divset(b4,len b4) &
         2 <= len b4 &
         b4 <= b5 &
         rng b5 = (rng b4) \/ {b1} &
         b3 is_bounded_on b2
   holds (Sum upper_volume(b3,b4)) - Sum upper_volume(b3,b5) <= ((sup rng b3) - inf rng b3) * delta b4;

:: INTEGRA3:th 11
theorem
for b1 being closed-interval Element of bool REAL
for b2 being Element of divs b1
for b3 being Element of REAL
for b4, b5 being Element of NAT
      st b4 in dom b2 & b5 in dom b2 & b4 <= b5 & b3 < (mid(b2,b4,b5)) . 1
   holds ex b6 being closed-interval Element of bool REAL st
      b3 = inf b6 &
       sup b6 = (mid(b2,b4,b5)) . len mid(b2,b4,b5) &
       len mid(b2,b4,b5) = (b5 - b4) + 1 &
       mid(b2,b4,b5) is DivisionPoint of b6;

:: INTEGRA3:th 12
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
for b4, b5 being Element of divs b2
      st b1 in divset(b4,len b4) &
         vol b2 <> 0 &
         b4 <= b5 &
         rng b5 = (rng b4) \/ {b1} &
         b3 is_bounded_on b2 &
         inf b2 < b1
   holds (Sum lower_volume(b3,b5)) - Sum lower_volume(b3,b4) <= ((sup rng b3) - inf rng b3) * delta b4;

:: INTEGRA3:th 13
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
for b4, b5 being Element of divs b2
      st b1 in divset(b4,len b4) &
         vol b2 <> 0 &
         b4 <= b5 &
         rng b5 = (rng b4) \/ {b1} &
         b3 is_bounded_on b2 &
         inf b2 < b1
   holds (Sum upper_volume(b3,b4)) - Sum upper_volume(b3,b5) <= ((sup rng b3) - inf rng b3) * delta b4;

:: INTEGRA3:th 14
theorem
for b1 being closed-interval Element of bool REAL
for b2, b3 being Element of divs b1
for b4 being Element of REAL
for b5, b6 being Element of NAT
      st b5 in dom b2 &
         b6 in dom b2 &
         b5 <= b6 &
         b2 <= b3 &
         b4 < (mid(b3,indx(b3,b2,b5),indx(b3,b2,b6))) . 1
   holds ex b7 being closed-interval Element of bool REAL st
      ex b8, b9 being Element of divs b7 st
         b4 = inf b7 &
          sup b7 = b9 . len b9 &
          sup b7 = b8 . len b8 &
          b8 <= b9 &
          b8 = mid(b2,b5,b6) &
          b9 = mid(b3,indx(b3,b2,b5),indx(b3,b2,b6));

:: INTEGRA3:th 15
theorem
for b1 being Element of REAL
for b2 being closed-interval Element of bool REAL
for b3 being Element of divs b2
      st b1 in rng b3
   holds b3 . 1 <= b1 & b1 <= b3 . len b3;

:: INTEGRA3:th 16
theorem
for b1 being FinSequence of REAL
for b2, b3, b4 being Element of NAT
      st b1 is increasing & b2 in dom b1 & b3 in dom b1 & b4 in dom b1 & b1 . b2 <= b1 . b4 & b1 . b4 <= b1 . b3
   holds b1 . b4 in rng mid(b1,b2,b3);

:: INTEGRA3:th 17
theorem
for b1 being Element of NAT
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being Element of divs b2
      st b3 is_bounded_on b2 & b1 in dom b4
   holds inf rng (b3 | divset(b4,b1)) <= sup rng b3;

:: INTEGRA3:th 18
theorem
for b1 being Element of NAT
for b2 being closed-interval Element of bool REAL
for b3 being Function-like quasi_total Relation of b2,REAL
for b4 being Element of divs b2
      st b3 is_bounded_on b2 & b1 in dom b4
   holds inf rng b3 <= sup rng (b3 | divset(b4,b1));

:: INTEGRA3:th 19
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_to_0 & vol b1 <> 0
   holds lower_sum(b2,b3) is convergent & lim lower_sum(b2,b3) = lower_integral b2;

:: INTEGRA3:th 20
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_to_0 & vol b1 <> 0
   holds upper_sum(b2,b3) is convergent & lim upper_sum(b2,b3) = upper_integral b2;