Article INTEGRA7, MML version 4.99.1005

:: INTEGRA7:th 1
theorem
for b1 being real set
for b2 being non empty set
for b3, b4 being Function-like quasi_total Relation of b2,REAL
      st rng b3 is bounded_above &
         rng b4 is bounded_above &
         (for b5 being set
               st b5 in b2
            holds abs ((b3 . b5) - (b4 . b5)) <= b1)
   holds (sup rng b3) - sup rng b4 <= b1 &
    (sup rng b4) - sup rng b3 <= b1;

:: INTEGRA7:th 2
theorem
for b1 being real set
for b2 being non empty set
for b3, b4 being Function-like quasi_total Relation of b2,REAL
      st rng b3 is bounded_below &
         rng b4 is bounded_below &
         (for b5 being set
               st b5 in b2
            holds abs ((b3 . b5) - (b4 . b5)) <= b1)
   holds (inf rng b3) - inf rng b4 <= b1 &
    (inf rng b4) - inf rng b3 <= b1;

:: INTEGRA7:th 3
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 | b1 is_bounded_on b1
   holds b2 is_bounded_on b1;

:: INTEGRA7:th 4
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st b3 in b1 & b2 | b1 is_differentiable_in b3
   holds b2 is_differentiable_in b3;

:: INTEGRA7:th 5
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 | b1 is_differentiable_on b1
   holds b2 is_differentiable_on b1;

:: INTEGRA7:th 6
theorem
for b1 being set
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_differentiable_on b1 & b3 is_differentiable_on b1
   holds b2 + b3 is_differentiable_on b1 & b2 - b3 is_differentiable_on b1 & b2 (#) b3 is_differentiable_on b1;

:: INTEGRA7:th 7
theorem
for b1 being real set
for b2 being set
for b3 being Function-like Relation of REAL,REAL
      st b3 is_differentiable_on b2
   holds b1 (#) b3 is_differentiable_on b2;

:: INTEGRA7:th 8
theorem
for b1 being set
for b2, b3 being Function-like Relation of REAL,REAL
      st (for b4 being set
               st b4 in b1
            holds b2 . b4 <> 0) &
         b3 is_differentiable_on b1 &
         b2 is_differentiable_on b1
   holds b3 / b2 is_differentiable_on b1;

:: INTEGRA7:th 9
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st (for b3 being set
               st b3 in b1
            holds b2 . b3 <> 0) &
         b2 is_differentiable_on b1
   holds b2 ^ is_differentiable_on b1;

:: INTEGRA7:th 10
theorem
for b1, b2 being real set
for b3 being set
for b4 being Function-like Relation of REAL,REAL
      st b1 <= b2 & ['b1,b2'] c= b3 & b4 is_differentiable_on b3 & b4 `| b3 is_integrable_on ['b1,b2'] & b4 `| b3 is_bounded_on ['b1,b2']
   holds b4 . b2 = (integral(b4 `| b3,b1,b2)) + (b4 . b1);

:: INTEGRA7:funcnot 1 => INTEGRA7:func 1
definition
  let a1 be set;
  let a2 be Function-like Relation of REAL,REAL;
  func IntegralFuncs(A2,A1) -> set means
    for b1 being set holds
          b1 in it
       iff
          ex b2 being Function-like Relation of REAL,REAL st
             b1 = b2 & b2 is_differentiable_on a1 & b2 `| a1 = a2 | a1;
end;

:: INTEGRA7:def 1
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
for b3 being set holds
      b3 = IntegralFuncs(b2,b1)
   iff
      for b4 being set holds
            b4 in b3
         iff
            ex b5 being Function-like Relation of REAL,REAL st
               b4 = b5 & b5 is_differentiable_on b1 & b5 `| b1 = b2 | b1;

:: INTEGRA7:prednot 1 => INTEGRA7:pred 1
definition
  let a1 be set;
  let a2, a3 be Function-like Relation of REAL,REAL;
  pred A2 is_integral_of A3,A1 means
    a2 in IntegralFuncs(a3,a1);
end;

:: INTEGRA7:dfs 2
definiens
  let a1 be set;
  let a2, a3 be Function-like Relation of REAL,REAL;
To prove
     a2 is_integral_of a3,a1
it is sufficient to prove
  thus a2 in IntegralFuncs(a3,a1);

:: INTEGRA7:def 2
theorem
for b1 being set
for b2, b3 being Function-like Relation of REAL,REAL holds
   b2 is_integral_of b3,b1
iff
   b2 in IntegralFuncs(b3,b1);

:: INTEGRA7:th 11
theorem
for b1 being set
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is_integral_of b3,b1
   holds b1 c= dom b2;

:: INTEGRA7:th 12
theorem
for b1 being set
for b2, b3, b4, b5 being Function-like Relation of REAL,REAL
      st b2 is_integral_of b3,b1 & b4 is_integral_of b5,b1
   holds b2 + b4 is_integral_of b3 + b5,b1 & b2 - b4 is_integral_of b3 - b5,b1;

:: INTEGRA7:th 13
theorem
for b1 being real set
for b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
      st b3 is_integral_of b4,b2
   holds b1 (#) b3 is_integral_of b1 (#) b4,b2;

:: INTEGRA7:th 14
theorem
for b1 being set
for b2, b3, b4, b5 being Function-like Relation of REAL,REAL
      st b2 is_integral_of b3,b1 & b4 is_integral_of b5,b1
   holds b2 (#) b4 is_integral_of (b3 (#) b4) + (b2 (#) b5),b1;

:: INTEGRA7:th 15
theorem
for b1 being set
for b2, b3, b4, b5 being Function-like Relation of REAL,REAL
      st (for b6 being set
               st b6 in b1
            holds b2 . b6 <> 0) &
         b3 is_integral_of b4,b1 &
         b2 is_integral_of b5,b1
   holds b3 / b2 is_integral_of ((b4 (#) b2) - (b3 (#) b5)) / (b2 (#) b2),b1;

:: INTEGRA7:th 16
theorem
for b1, b2 being real set
for b3, b4 being Function-like Relation of REAL,REAL
      st b1 <= b2 &
         ['b1,b2'] c= dom b3 &
         b3 is_continuous_on ['b1,b2'] &
         ].b1,b2.[ c= dom b4 &
         (for b5 being real set
               st b5 in ].b1,b2.[
            holds b4 . b5 = (integral(b3,b1,b5)) + (b4 . b1))
   holds b4 is_integral_of b3,].b1,b2.[;

:: INTEGRA7:th 17
theorem
for b1, b2 being real set
for b3, b4 being Function-like Relation of REAL,REAL
for b5, b6 being real set
      st b3 is_continuous_on [.b1,b2.] & b5 in ].b1,b2.[ & b6 in ].b1,b2.[ & b4 is_integral_of b3,].b1,b2.[
   holds b4 . b5 = (integral(b3,b6,b5)) + (b4 . b6);

:: INTEGRA7:th 18
theorem
for b1, b2 being real set
for b3 being set
for b4, b5 being Function-like Relation of REAL,REAL
      st b1 <= b2 & ['b1,b2'] c= b3 & b4 is_integral_of b5,b3 & b5 is_integrable_on ['b1,b2'] & b5 is_bounded_on ['b1,b2']
   holds b4 . b2 = (integral(b5,b1,b2)) + (b4 . b1);

:: INTEGRA7:th 19
theorem
for b1, b2 being real set
for b3 being set
for b4 being Function-like Relation of REAL,REAL
      st b1 <= b2 & [.b1,b2.] c= b3 & b4 is_continuous_on b3
   holds b4 is_continuous_on ['b1,b2'] & b4 is_integrable_on ['b1,b2'] & b4 is_bounded_on ['b1,b2'];

:: INTEGRA7:th 20
theorem
for b1, b2 being real set
for b3 being set
for b4, b5 being Function-like Relation of REAL,REAL
      st b1 <= b2 & [.b1,b2.] c= b3 & b4 is_continuous_on b3 & b5 is_integral_of b4,b3
   holds b5 . b2 = (integral(b4,b1,b2)) + (b5 . b1);

:: INTEGRA7:th 21
theorem
for b1, b2 being real set
for b3 being set
for b4, b5, b6, b7 being Function-like Relation of REAL,REAL
      st b1 <= b2 & ['b1,b2'] c= b3 & b4 is_integrable_on ['b1,b2'] & b5 is_integrable_on ['b1,b2'] & b4 is_bounded_on ['b1,b2'] & b5 is_bounded_on ['b1,b2'] & b3 c= dom b4 & b3 c= dom b5 & b6 is_integral_of b4,b3 & b7 is_integral_of b5,b3
   holds ((b6 . b2) * (b7 . b2)) - ((b6 . b1) * (b7 . b1)) = (integral(b4 (#) b7,b1,b2)) + integral(b6 (#) b5,b1,b2);

:: INTEGRA7:th 22
theorem
for b1, b2 being real set
for b3 being set
for b4, b5, b6, b7 being Function-like Relation of REAL,REAL
      st b1 <= b2 & [.b1,b2.] c= b3 & b3 c= dom b4 & b3 c= dom b5 & b4 is_continuous_on b3 & b5 is_continuous_on b3 & b6 is_integral_of b4,b3 & b7 is_integral_of b5,b3
   holds ((b6 . b2) * (b7 . b2)) - ((b6 . b1) * (b7 . b1)) = (integral(b4 (#) b7,b1,b2)) + integral(b6 (#) b5,b1,b2);

:: INTEGRA7:th 23
theorem
sin is_integral_of cos,REAL;

:: INTEGRA7:th 24
theorem
for b1, b2 being real set holds
(sin . b1) - (sin . b2) = integral(cos,b2,b1);

:: INTEGRA7:th 25
theorem
(- 1) (#) cos is_integral_of sin,REAL;

:: INTEGRA7:th 26
theorem
for b1, b2 being real set holds
(cos . b1) - (cos . b2) = integral(sin,b1,b2);

:: INTEGRA7:th 27
theorem
exp_R is_integral_of exp_R,REAL;

:: INTEGRA7:th 28
theorem
for b1, b2 being real set holds
(exp_R . b1) - (exp_R . b2) = integral(exp_R,b2,b1);

:: INTEGRA7:th 29
theorem
for b1 being Element of NAT holds
   #Z (b1 + 1) is_integral_of (b1 + 1) (#) #Z b1,REAL;

:: INTEGRA7:th 30
theorem
for b1, b2 being real set
for b3 being Element of NAT holds
   ((#Z (b3 + 1)) . b1) - ((#Z (b3 + 1)) . b2) = integral((b3 + 1) (#) #Z b3,b2,b1);

:: INTEGRA7:th 31
theorem
for b1, b2 being real set
for b3 being Functional_Sequence of REAL,REAL
for b4 being Function-like quasi_total Relation of NAT,REAL
      st b1 < b2 &
         (for b5 being Element of NAT holds
            b3 . b5 is_integrable_on ['b1,b2'] & b3 . b5 is_bounded_on ['b1,b2'] & b4 . b5 = integral(b3 . b5,b1,b2)) &
         b3 is_unif_conv_on ['b1,b2']
   holds lim(b3,['b1,b2']) is_bounded_on ['b1,b2'] &
    lim(b3,['b1,b2']) is_integrable_on ['b1,b2'] &
    b4 is convergent &
    lim b4 = integral(lim(b3,['b1,b2']),b1,b2);