Article MESFUNC6, MML version 4.99.1005

:: MESFUNC6:th 1
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
   |.R_EAL b2.| = R_EAL abs b2;

:: MESFUNC6:th 2
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL
for b5 being Element of REAL
      st dom b4 in b2 &
         (for b6 being set
               st b6 in dom b4
            holds b4 . b6 = b5)
   holds b4 is_simple_func_in b2;

:: MESFUNC6:th 3
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being real set
for b4 being set holds
      b4 in less_dom(b2,b3)
   iff
      b4 in dom b2 &
       (ex b5 being Element of REAL st
          b5 = b2 . b4 & b5 < b3);

:: MESFUNC6:th 4
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being real set
for b4 being set holds
      b4 in less_eq_dom(b2,b3)
   iff
      b4 in dom b2 &
       (ex b5 being Element of REAL st
          b5 = b2 . b4 & b5 <= b3);

:: MESFUNC6:th 5
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
for b4 being set holds
      b4 in great_dom(b2,b3)
   iff
      b4 in dom b2 &
       (ex b5 being Element of REAL st
          b5 = b2 . b4 & b3 < b5);

:: MESFUNC6:th 6
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
for b4 being set holds
      b4 in great_eq_dom(b2,b3)
   iff
      b4 in dom b2 &
       (ex b5 being Element of REAL st
          b5 = b2 . b4 & b3 <= b5);

:: MESFUNC6:th 7
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
for b4 being set holds
      b4 in eq_dom(b2,b3)
   iff
      b4 in dom b2 &
       (ex b5 being Element of REAL st
          b5 = b2 . b4 & b3 = b5);

:: MESFUNC6:th 8
theorem
for b1 being non empty set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being Function-like quasi_total Relation of NAT,b3
for b5 being Function-like Relation of b1,REAL
for b6 being Element of REAL
      st for b7 being natural set holds
           b4 . b7 = b2 /\ great_dom(b5,b6 - (1 / (b7 + 1)))
   holds b2 /\ great_eq_dom(b5,b6) = meet rng b4;

:: MESFUNC6:th 9
theorem
for b1 being non empty set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being Function-like quasi_total Relation of NAT,b3
for b5 being Function-like Relation of b1,REAL
for b6 being Element of REAL
      st for b7 being natural set holds
           b4 . b7 = b2 /\ less_dom(b5,b6 + (1 / (b7 + 1)))
   holds b2 /\ less_eq_dom(b5,b6) = meet rng b4;

:: MESFUNC6:th 10
theorem
for b1 being non empty set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being Function-like quasi_total Relation of NAT,b3
for b5 being Function-like Relation of b1,REAL
for b6 being Element of REAL
      st for b7 being natural set holds
           b4 . b7 = b2 /\ less_eq_dom(b5,b6 - (1 / (b7 + 1)))
   holds b2 /\ less_dom(b5,b6) = union rng b4;

:: MESFUNC6:th 11
theorem
for b1 being non empty set
for b2 being set
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being Function-like quasi_total Relation of NAT,b3
for b5 being Function-like Relation of b1,REAL
for b6 being Element of REAL
      st for b7 being natural set holds
           b4 . b7 = b2 /\ great_eq_dom(b5,b6 + (1 / (b7 + 1)))
   holds b2 /\ great_dom(b5,b6) = union rng b4;

:: MESFUNC6:prednot 1 => MESFUNC6:pred 1
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like Relation of a1,REAL;
  let a4 be Element of a2;
  pred A3 is_measurable_on A4 means
    R_EAL a3 is_measurable_on a4;
end;

:: MESFUNC6:dfs 1
definiens
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like Relation of a1,REAL;
  let a4 be Element of a2;
To prove
     a3 is_measurable_on a4
it is sufficient to prove
  thus R_EAL a3 is_measurable_on a4;

:: MESFUNC6:def 6
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2 holds
      b3 is_measurable_on b4
   iff
      R_EAL b3 is_measurable_on b4;

:: MESFUNC6:th 12
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2 holds
      b3 is_measurable_on b4
   iff
      for b5 being real set holds
         b4 /\ less_dom(b3,b5) in b2;

:: MESFUNC6:th 13
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2
      st b4 c= dom b3
   holds    b3 is_measurable_on b4
   iff
      for b5 being real set holds
         b4 /\ great_eq_dom(b3,b5) in b2;

:: MESFUNC6:th 14
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2 holds
      b3 is_measurable_on b4
   iff
      for b5 being real set holds
         b4 /\ less_eq_dom(b3,b5) in b2;

:: MESFUNC6:th 15
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2
      st b4 c= dom b3
   holds    b3 is_measurable_on b4
   iff
      for b5 being real set holds
         b4 /\ great_dom(b3,b5) in b2;

:: MESFUNC6:th 16
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4, b5 being Element of b2
      st b4 c= b5 & b3 is_measurable_on b5
   holds b3 is_measurable_on b4;

:: MESFUNC6:th 17
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4, b5 being Element of b2
      st b3 is_measurable_on b4 & b3 is_measurable_on b5
   holds b3 is_measurable_on b4 \/ b5;

:: MESFUNC6:th 18
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2
for b5, b6 being Element of REAL
      st b3 is_measurable_on b4 & b4 c= dom b3
   holds (b4 /\ great_dom(b3,b5)) /\ less_dom(b3,b6) in b2;

:: MESFUNC6:th 19
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,REAL
for b5 being Element of b2
for b6 being Element of REAL
      st b3 is_measurable_on b5 & b4 is_measurable_on b5 & b5 c= dom b4
   holds (b5 /\ less_dom(b3,b6)) /\ great_dom(b4,b6) in b2;

:: MESFUNC6:th 20
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL holds
   R_EAL (b3 (#) b2) = b3 (#) R_EAL b2;

:: MESFUNC6:th 21
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2
for b5 being Element of REAL
      st b3 is_measurable_on b4 & b4 c= dom b3
   holds b5 (#) b3 is_measurable_on b4;

:: MESFUNC6:th 22
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
   R_EAL b2 is real-valued;

:: MESFUNC6:th 23
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL holds
R_EAL (b2 + b3) = (R_EAL b2) + R_EAL b3 &
 R_EAL (b2 - b3) = (R_EAL b2) - R_EAL b3 &
 dom R_EAL (b2 + b3) = (dom R_EAL b2) /\ dom R_EAL b3 &
 dom R_EAL (b2 - b3) = (dom R_EAL b2) /\ dom R_EAL b3 &
 dom R_EAL (b2 + b3) = (dom b2) /\ dom b3 &
 dom R_EAL (b2 - b3) = (dom b2) /\ dom b3;

:: MESFUNC6:th 24
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,REAL
for b5 being Element of b2
for b6 being Element of REAL
for b7 being Function-like quasi_total Relation of RAT,b2
      st for b8 being rational set holds
           b7 . b8 = (b5 /\ less_dom(b3,b8)) /\ (b5 /\ less_dom(b4,b6 - b8))
   holds b5 /\ less_dom(b3 + b4,b6) = union rng b7;

:: MESFUNC6:th 25
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,REAL
for b5 being Element of b2
for b6 being Element of REAL
      st b3 is_measurable_on b5 & b4 is_measurable_on b5
   holds ex b7 being Function-like quasi_total Relation of RAT,b2 st
      for b8 being rational set holds
         b7 . b8 = (b5 /\ less_dom(b3,b8)) /\ (b5 /\ less_dom(b4,b6 - b8));

:: MESFUNC6:th 26
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,REAL
for b5 being Element of b2
      st b3 is_measurable_on b5 & b4 is_measurable_on b5
   holds b3 + b4 is_measurable_on b5;

:: MESFUNC6:th 27
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL holds
(R_EAL b2) - R_EAL b3 = (R_EAL b2) + R_EAL - b3;

:: MESFUNC6:th 28
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
   - R_EAL b2 = R_EAL ((- 1) (#) b2) &
    - R_EAL b2 = R_EAL - b2;

:: MESFUNC6:th 29
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,REAL
for b5 being Element of b2
      st b3 is_measurable_on b5 & b4 is_measurable_on b5 & b5 c= dom b4
   holds b3 - b4 is_measurable_on b5;

:: MESFUNC6:th 30
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
   max+ R_EAL b2 = max+ b2 & max- R_EAL b2 = max- b2;

:: MESFUNC6:th 31
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of b1 holds
   0 <= (max+ b2) . b3;

:: MESFUNC6:th 32
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of b1 holds
   0 <= (max- b2) . b3;

:: MESFUNC6:th 33
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
   max- b2 = max+ - b2;

:: MESFUNC6:th 34
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
      st b3 in dom b2 & 0 < (max+ b2) . b3
   holds (max- b2) . b3 = 0;

:: MESFUNC6:th 35
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
      st b3 in dom b2 & 0 < (max- b2) . b3
   holds (max+ b2) . b3 = 0;

:: MESFUNC6:th 36
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
   dom b2 = dom ((max+ b2) - max- b2) &
    dom b2 = dom ((max+ b2) + max- b2);

:: MESFUNC6:th 37
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
      st b3 in dom b2
   holds ((max+ b2) . b3 = b2 . b3 or (max+ b2) . b3 = 0) &
    ((max- b2) . b3 = - (b2 . b3) or (max- b2) . b3 = 0);

:: MESFUNC6:th 38
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
      st b3 in dom b2 & (max+ b2) . b3 = b2 . b3
   holds (max- b2) . b3 = 0;

:: MESFUNC6:th 39
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
      st b3 in dom b2 & (max+ b2) . b3 = 0
   holds (max- b2) . b3 = - (b2 . b3);

:: MESFUNC6:th 40
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
      st b3 in dom b2 & (max- b2) . b3 = - (b2 . b3)
   holds (max+ b2) . b3 = 0;

:: MESFUNC6:th 41
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
      st b3 in dom b2 & (max- b2) . b3 = 0
   holds (max+ b2) . b3 = b2 . b3;

:: MESFUNC6:th 42
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
   b2 = (max+ b2) - max- b2;

:: MESFUNC6:th 43
theorem
for b1 being Element of REAL holds
   |.b1.| = |.R_EAL b1.|;

:: MESFUNC6:th 44
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
   R_EAL abs b2 = |.R_EAL b2.|;

:: MESFUNC6:th 45
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
   abs b2 = (max+ b2) + max- b2;

:: MESFUNC6:th 46
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2
      st b3 is_measurable_on b4
   holds max+ b3 is_measurable_on b4;

:: MESFUNC6:th 47
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2
      st b3 is_measurable_on b4 & b4 c= dom b3
   holds max- b3 is_measurable_on b4;

:: MESFUNC6:th 48
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2
      st b3 is_measurable_on b4 & b4 c= dom b3
   holds abs b3 is_measurable_on b4;

:: MESFUNC6:prednot 2 => MESFUNC6:pred 2
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like Relation of a1,REAL;
  pred A3 is_simple_func_in A2 means
    ex b1 being disjoint_valued FinSequence of a2 st
       dom a3 = union rng b1 &
        (for b2 being natural set
        for b3, b4 being Element of a1
              st b2 in dom b1 & b3 in b1 . b2 & b4 in b1 . b2
           holds a3 . b3 = a3 . b4);
end;

:: MESFUNC6:dfs 2
definiens
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be Function-like Relation of a1,REAL;
To prove
     a3 is_simple_func_in a2
it is sufficient to prove
  thus ex b1 being disjoint_valued FinSequence of a2 st
       dom a3 = union rng b1 &
        (for b2 being natural set
        for b3, b4 being Element of a1
              st b2 in dom b1 & b3 in b1 . b2 & b4 in b1 . b2
           holds a3 . b3 = a3 . b4);

:: MESFUNC6:def 7
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL holds
      b3 is_simple_func_in b2
   iff
      ex b4 being disjoint_valued FinSequence of b2 st
         dom b3 = union rng b4 &
          (for b5 being natural set
          for b6, b7 being Element of b1
                st b5 in dom b4 & b6 in b4 . b5 & b7 in b4 . b5
             holds b3 . b6 = b3 . b7);

:: MESFUNC6:th 49
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL holds
      b3 is_simple_func_in b2
   iff
      R_EAL b3 is_simple_func_in b2;

:: MESFUNC6:th 50
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2
      st b3 is_simple_func_in b2
   holds b3 is_measurable_on b4;

:: MESFUNC6:th 51
theorem
for b1 being set
for b2 being Function-like Relation of b1,REAL holds
      b2 is nonnegative
   iff
      for b3 being set holds
         0 <= b2 . b3;

:: MESFUNC6:th 52
theorem
for b1 being set
for b2 being Function-like Relation of b1,REAL
      st for b3 being set
              st b3 in dom b2
           holds 0 <= b2 . b3
   holds b2 is nonnegative;

:: MESFUNC6:th 53
theorem
for b1 being set
for b2 being Function-like Relation of b1,REAL holds
      b2 is nonpositive
   iff
      for b3 being set holds
         b2 . b3 <= 0;

:: MESFUNC6:th 54
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
      st for b3 being set
              st b3 in dom b2
           holds b2 . b3 <= 0
   holds b2 is nonpositive;

:: MESFUNC6:th 55
theorem
for b1 being non empty set
for b2 being set
for b3 being Function-like Relation of b1,REAL
      st b3 is nonnegative
   holds b3 | b2 is nonnegative;

:: MESFUNC6:th 56
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL
      st b2 is nonnegative & b3 is nonnegative
   holds b2 + b3 is nonnegative;

:: MESFUNC6:th 57
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
      st b2 is nonnegative
   holds (0 <= b3 implies b3 (#) b2 is nonnegative) & (b3 <= 0 implies b3 (#) b2 is nonpositive);

:: MESFUNC6:th 58
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL
      st for b4 being set
              st b4 in (dom b2) /\ dom b3
           holds b3 . b4 <= b2 . b4
   holds b2 - b3 is nonnegative;

:: MESFUNC6:th 59
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,REAL
      st b2 is nonnegative & b3 is nonnegative & b4 is nonnegative
   holds (b2 + b3) + b4 is nonnegative;

:: MESFUNC6:th 60
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,REAL
for b5 being set
      st b5 in dom ((b2 + b3) + b4)
   holds ((b2 + b3) + b4) . b5 = ((b2 . b5) + (b3 . b5)) + (b4 . b5);

:: MESFUNC6:th 61
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
   max+ b2 is nonnegative & max- b2 is nonnegative;

:: MESFUNC6:th 62
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL holds
dom ((max+ (b2 + b3)) + max- b2) = (dom b2) /\ dom b3 &
 dom ((max- (b2 + b3)) + max+ b2) = (dom b2) /\ dom b3 &
 dom (((max+ (b2 + b3)) + max- b2) + max- b3) = (dom b2) /\ dom b3 &
 dom (((max- (b2 + b3)) + max+ b2) + max+ b3) = (dom b2) /\ dom b3 &
 (max+ (b2 + b3)) + max- b2 is nonnegative &
 (max- (b2 + b3)) + max+ b2 is nonnegative;

:: MESFUNC6:th 63
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL holds
((max+ (b2 + b3)) + max- b2) + max- b3 = ((max- (b2 + b3)) + max+ b2) + max+ b3;

:: MESFUNC6:th 64
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
      st 0 <= b3
   holds max+ (b3 (#) b2) = b3 (#) max+ b2 &
    max- (b3 (#) b2) = b3 (#) max- b2;

:: MESFUNC6:th 65
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
      st 0 <= b3
   holds max+ ((- b3) (#) b2) = b3 (#) max- b2 &
    max- ((- b3) (#) b2) = b3 (#) max+ b2;

:: MESFUNC6:th 66
theorem
for b1 being non empty set
for b2 being set
for b3 being Function-like Relation of b1,REAL holds
   max+ (b3 | b2) = (max+ b3) | b2 & max- (b3 | b2) = (max- b3) | b2;

:: MESFUNC6:th 67
theorem
for b1 being non empty set
for b2 being set
for b3, b4 being Function-like Relation of b1,REAL
      st b2 c= dom (b3 + b4)
   holds dom ((b3 + b4) | b2) = b2 &
    dom ((b3 | b2) + (b4 | b2)) = b2 &
    (b3 + b4) | b2 = (b3 | b2) + (b4 | b2);

:: MESFUNC6:th 68
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL holds
   eq_dom(b2,b3) = b2 " {b3};

:: MESFUNC6:th 69
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2
for b5, b6 being Element of REAL
      st b3 is_measurable_on b4 & b4 c= dom b3
   holds (b4 /\ great_eq_dom(b3,b5)) /\ less_dom(b3,b6) in b2;

:: MESFUNC6:th 70
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of b2
      st b3 is_simple_func_in b2
   holds b3 | b4 is_simple_func_in b2;

:: MESFUNC6:th 71
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
      st b3 is_simple_func_in b2
   holds dom b3 is Element of b2;

:: MESFUNC6:th 72
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,REAL
      st b3 is_simple_func_in b2 & b4 is_simple_func_in b2
   holds b3 + b4 is_simple_func_in b2;

:: MESFUNC6:th 73
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4 being Element of REAL
      st b3 is_simple_func_in b2
   holds b4 (#) b3 is_simple_func_in b2;

:: MESFUNC6:th 74
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL
      st for b4 being set
              st b4 in dom (b2 - b3)
           holds b3 . b4 <= b2 . b4
   holds b2 - b3 is nonnegative;

:: MESFUNC6:th 75
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Element of b2
for b4 being Element of REAL holds
   ex b5 being Function-like Relation of b1,REAL st
      b5 is_simple_func_in b2 &
       dom b5 = b3 &
       (for b6 being set
             st b6 in b3
          holds b5 . b6 = b4);

:: MESFUNC6:th 76
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4, b5 being Element of b2
      st b3 is_measurable_on b4 & b5 = (dom b3) /\ b4
   holds b3 | b4 is_measurable_on b5;

:: MESFUNC6:th 77
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,REAL
for b5 being Element of b2
      st b5 c= dom b3 & b3 is_measurable_on b5 & b4 is_measurable_on b5
   holds (max+ (b3 + b4)) + max- b3 is_measurable_on b5;

:: MESFUNC6:th 78
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,REAL
for b5 being Element of b2
      st b5 c= (dom b3) /\ dom b4 & b3 is_measurable_on b5 & b4 is_measurable_on b5
   holds (max- (b3 + b4)) + max+ b3 is_measurable_on b5;

:: MESFUNC6:th 79
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,REAL
      st dom b3 in b2 & dom b4 in b2
   holds dom (b3 + b4) in b2;

:: MESFUNC6:th 80
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
for b4, b5 being Element of b2
      st dom b3 = b4
   holds    b3 is_measurable_on b5
   iff
      b3 is_measurable_on b4 /\ b5;

:: MESFUNC6:th 81
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,REAL
   st ex b4 being Element of b2 st
        dom b3 = b4
for b4 being Element of REAL
for b5 being Element of b2
      st b3 is_measurable_on b5
   holds b4 (#) b3 is_measurable_on b5;

:: MESFUNC6:funcnot 1 => MESFUNC6:func 1
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
  let a4 be Function-like Relation of a1,REAL;
  func Integral(A3,A4) -> Element of ExtREAL equals
    Integral(a3,R_EAL a4);
end;

:: MESFUNC6:def 8
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL holds
   Integral(b3,b4) = Integral(b3,R_EAL b4);

:: MESFUNC6:th 82
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
      st (ex b5 being Element of b2 st
            b5 = dom b4 & b4 is_measurable_on b5) &
         b4 is nonnegative
   holds Integral(b3,b4) = integral+(b3,R_EAL b4);

:: MESFUNC6:th 83
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
      st b4 is_simple_func_in b2 & b4 is nonnegative
   holds Integral(b3,b4) = integral+(b3,R_EAL b4) & Integral(b3,b4) = integral'(b3,R_EAL b4);

:: MESFUNC6:th 84
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
      st (ex b5 being Element of b2 st
            b5 = dom b4 & b4 is_measurable_on b5) &
         b4 is nonnegative
   holds 0 <= Integral(b3,b4);

:: MESFUNC6:th 85
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
for b5, b6 being Element of b2
      st (ex b7 being Element of b2 st
            b7 = dom b4 & b4 is_measurable_on b7) &
         b4 is nonnegative &
         b5 misses b6
   holds Integral(b3,b4 | (b5 \/ b6)) = (Integral(b3,b4 | b5)) + Integral(b3,b4 | b6);

:: MESFUNC6:th 86
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
for b5 being Element of b2
      st (ex b6 being Element of b2 st
            b6 = dom b4 & b4 is_measurable_on b6) &
         b4 is nonnegative
   holds 0 <= Integral(b3,b4 | b5);

:: MESFUNC6:th 87
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
for b5, b6 being Element of b2
      st (ex b7 being Element of b2 st
            b7 = dom b4 & b4 is_measurable_on b7) &
         b4 is nonnegative &
         b5 c= b6
   holds Integral(b3,b4 | b5) <= Integral(b3,b4 | b6);

:: MESFUNC6:th 88
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
for b5 being Element of b2
      st (ex b6 being Element of b2 st
            b6 = dom b4 & b4 is_measurable_on b6) &
         b3 . b5 = 0
   holds Integral(b3,b4 | b5) = 0;

:: MESFUNC6:th 89
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
for b5, b6 being Element of b2
      st b5 = dom b4 & b4 is_measurable_on b5 & b3 . b6 = 0
   holds Integral(b3,b4 | (b5 \ b6)) = Integral(b3,b4);

:: MESFUNC6:prednot 3 => MESFUNC6:pred 3
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
  let a4 be Function-like Relation of a1,REAL;
  pred A4 is_integrable_on A3 means
    R_EAL a4 is_integrable_on a3;
end;

:: MESFUNC6:dfs 4
definiens
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
  let a4 be Function-like Relation of a1,REAL;
To prove
     a4 is_integrable_on a3
it is sufficient to prove
  thus R_EAL a4 is_integrable_on a3;

:: MESFUNC6:def 9
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL holds
      b4 is_integrable_on b3
   iff
      R_EAL b4 is_integrable_on b3;

:: MESFUNC6:th 90
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
      st b4 is_integrable_on b3
   holds -infty < Integral(b3,b4) & Integral(b3,b4) < +infty;

:: MESFUNC6:th 91
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
for b5 being Element of b2
      st b4 is_integrable_on b3
   holds b4 | b5 is_integrable_on b3;

:: MESFUNC6:th 92
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
for b5, b6 being Element of b2
      st b4 is_integrable_on b3 & b5 misses b6
   holds Integral(b3,b4 | (b5 \/ b6)) = (Integral(b3,b4 | b5)) + Integral(b3,b4 | b6);

:: MESFUNC6:th 93
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
for b5, b6 being Element of b2
      st b4 is_integrable_on b3 & b5 = (dom b4) \ b6
   holds b4 | b6 is_integrable_on b3 &
    Integral(b3,b4) = (Integral(b3,b4 | b6)) + Integral(b3,b4 | b5);

:: MESFUNC6:th 94
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
      st ex b5 being Element of b2 st
           b5 = dom b4 & b4 is_measurable_on b5
   holds    b4 is_integrable_on b3
   iff
      abs b4 is_integrable_on b3;

:: MESFUNC6:th 95
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
      st b4 is_integrable_on b3
   holds |.Integral(b3,b4).| <= Integral(b3,abs b4);

:: MESFUNC6:th 96
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,REAL
      st (ex b6 being Element of b2 st
            b6 = dom b4 & b4 is_measurable_on b6) &
         dom b4 = dom b5 &
         b5 is_integrable_on b3 &
         (for b6 being Element of b1
               st b6 in dom b4
            holds abs (b4 . b6) <= b5 . b6)
   holds b4 is_integrable_on b3 & Integral(b3,abs b4) <= Integral(b3,b5);

:: MESFUNC6:th 97
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
for b5 being Element of REAL
      st dom b4 in b2 &
         0 <= b5 &
         (for b6 being set
               st b6 in dom b4
            holds b4 . b6 = b5)
   holds Integral(b3,b4) = (R_EAL b5) * (b3 . dom b4);

:: MESFUNC6:th 98
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,REAL
      st b4 is_integrable_on b3 & b5 is_integrable_on b3 & b4 is nonnegative & b5 is nonnegative
   holds b4 + b5 is_integrable_on b3;

:: MESFUNC6:th 99
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,REAL
      st b4 is_integrable_on b3 & b5 is_integrable_on b3
   holds dom (b4 + b5) in b2;

:: MESFUNC6:th 100
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,REAL
      st b4 is_integrable_on b3 & b5 is_integrable_on b3
   holds b4 + b5 is_integrable_on b3;

:: MESFUNC6:th 101
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,REAL
      st b4 is_integrable_on b3 & b5 is_integrable_on b3
   holds ex b6 being Element of b2 st
      b6 = (dom b4) /\ dom b5 &
       Integral(b3,b4 + b5) = (Integral(b3,b4 | b6)) + Integral(b3,b5 | b6);

:: MESFUNC6:th 102
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
for b5 being Element of REAL
      st b4 is_integrable_on b3
   holds b5 (#) b4 is_integrable_on b3 &
    Integral(b3,b5 (#) b4) = (R_EAL b5) * Integral(b3,b4);

:: MESFUNC6:funcnot 2 => MESFUNC6:func 2
definition
  let a1 be non empty set;
  let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
  let a3 be sigma_Measure of a2;
  let a4 be Function-like Relation of a1,REAL;
  let a5 be Element of a2;
  func Integral_on(A3,A5,A4) -> Element of ExtREAL equals
    Integral(a3,a4 | a5);
end;

:: MESFUNC6:def 10
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
for b5 being Element of b2 holds
   Integral_on(b3,b5,b4) = Integral(b3,b4 | b5);

:: MESFUNC6:th 103
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,REAL
for b6 being Element of b2
      st b4 is_integrable_on b3 & b5 is_integrable_on b3 & b6 c= dom (b4 + b5)
   holds b4 + b5 is_integrable_on b3 &
    Integral_on(b3,b6,b4 + b5) = (Integral_on(b3,b6,b4)) + Integral_on(b3,b6,b5);

:: MESFUNC6:th 104
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like Relation of b1,REAL
for b5 being Element of REAL
for b6 being Element of b2
      st b4 is_integrable_on b3 & b4 is_measurable_on b6
   holds b4 | b6 is_integrable_on b3 &
    Integral_on(b3,b6,b5 (#) b4) = (R_EAL b5) * Integral_on(b3,b6,b4);