Article MESFUNC5, MML version 4.99.1005

:: MESFUNC5:th 1
theorem
for b1, b2 being Element of ExtREAL holds
|.b1 - b2.| = |.b2 - b1.|;

:: MESFUNC5:th 2
theorem
for b1, b2 being Element of ExtREAL holds
b2 - b1 <= |.b1 - b2.|;

:: MESFUNC5:th 3
theorem
for b1, b2 being Element of ExtREAL
for b3 being real set
      st |.b1 - b2.| < b3 & (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty)
   holds b1 <> +infty & b1 <> -infty & b2 <> +infty & b2 <> -infty;

:: MESFUNC5:th 4
theorem
for b1, b2 being Element of ExtREAL
      st for b3 being real set
              st 0 < b3
           holds b1 < b2 + R_EAL b3
   holds b1 <= b2;

:: MESFUNC5:th 5
theorem
for b1, b2, b3 being Element of ExtREAL
      st b3 <> -infty & b3 <> +infty & b1 < b2
   holds b1 + b3 < b2 + b3;

:: MESFUNC5:th 6
theorem
for b1, b2, b3 being Element of ExtREAL
      st b3 <> -infty & b3 <> +infty & b1 < b2
   holds b1 - b3 < b2 - b3;

:: MESFUNC5:th 7
theorem
for b1, b2 being real set holds
(R_EAL b1) + R_EAL b2 = b1 + b2 & - R_EAL b1 = - b1;

:: MESFUNC5:th 8
theorem
for b1 being natural set
for b2 being Element of ExtREAL
      st 0 <= b2 & b2 < b1
   holds ex b3 being natural set st
      1 <= b3 & b3 <= (2 |^ b1) * b1 & (b3 - 1) / (2 |^ b1) <= b2 & b2 < b3 / (2 |^ b1);

:: MESFUNC5:th 9
theorem
for b1, b2 being natural set
for b3 being Element of ExtREAL
      st 1 <= b2 & b2 <= (2 |^ b1) * b1 & b1 <= b3 & (b2 - 1) / (2 |^ b1) <= b3
   holds b2 / (2 |^ b1) <= b3;

:: MESFUNC5:th 10
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st -infty < b3 & b1 < b2 & b3 < b4
   holds b1 + b3 < b2 + b4;

:: MESFUNC5:th 11
theorem
for b1, b2, b3 being Element of ExtREAL
      st 0 <= b3
   holds b3 * max(b1,b2) = max(b3 * b1,b3 * b2) &
    b3 * min(b1,b2) = min(b3 * b1,b3 * b2);

:: MESFUNC5:th 12
theorem
for b1, b2, b3 being Element of ExtREAL
      st b3 <= 0
   holds b3 * min(b1,b2) = max(b3 * b1,b3 * b2) &
    b3 * max(b1,b2) = min(b3 * b1,b3 * b2);

:: MESFUNC5:th 13
theorem
for b1, b2, b3 being Element of ExtREAL
      st 0 <= b1 & 0 <= b3 & b3 + b1 <= b2
   holds b3 <= b2;

:: MESFUNC5:attrnot 1 => MESFUNC5:attr 1
definition
  let a1 be set;
  attr a1 is nonpositive means
    for b1 being Element of ExtREAL
          st b1 in a1
       holds b1 <= 0;
end;

:: MESFUNC5:dfs 1
definiens
  let a1 be set;
To prove
     a1 is nonpositive
it is sufficient to prove
  thus for b1 being Element of ExtREAL
          st b1 in a1
       holds b1 <= 0;

:: MESFUNC5:def 1
theorem
for b1 being set holds
      b1 is nonpositive
   iff
      for b2 being Element of ExtREAL
            st b2 in b1
         holds b2 <= 0;

:: MESFUNC5:attrnot 2 => MESFUNC5:attr 2
definition
  let a1 be Relation-like set;
  attr a1 is nonpositive means
    proj2 a1 is nonpositive;
end;

:: MESFUNC5:dfs 2
definiens
  let a1 be Relation-like set;
To prove
     a1 is nonpositive
it is sufficient to prove
  thus proj2 a1 is nonpositive;

:: MESFUNC5:def 2
theorem
for b1 being Relation-like set holds
      b1 is nonpositive
   iff
      proj2 b1 is nonpositive;

:: MESFUNC5:th 14
theorem
for b1 being set
for b2 being Function-like Relation of b1,ExtREAL holds
      b2 is nonpositive
   iff
      for b3 being set holds
         b2 . b3 <= 0.;

:: MESFUNC5:th 15
theorem
for b1 being set
for b2 being Function-like Relation of b1,ExtREAL
      st for b3 being set
              st b3 in dom b2
           holds b2 . b3 <= 0.
   holds b2 is nonpositive;

:: MESFUNC5:attrnot 3 => MESFUNC5:attr 3
definition
  let a1 be Relation-like set;
  attr a1 is without-infty means
    not -infty in proj2 a1;
end;

:: MESFUNC5:dfs 3
definiens
  let a1 be Relation-like set;
To prove
     a1 is without-infty
it is sufficient to prove
  thus not -infty in proj2 a1;

:: MESFUNC5:def 3
theorem
for b1 being Relation-like set holds
      b1 is without-infty
   iff
      not -infty in proj2 b1;

:: MESFUNC5:attrnot 4 => MESFUNC5:attr 4
definition
  let a1 be Relation-like set;
  attr a1 is without+infty means
    not +infty in proj2 a1;
end;

:: MESFUNC5:dfs 4
definiens
  let a1 be Relation-like set;
To prove
     a1 is without+infty
it is sufficient to prove
  thus not +infty in proj2 a1;

:: MESFUNC5:def 4
theorem
for b1 being Relation-like set holds
      b1 is without+infty
   iff
      not +infty in proj2 b1;

:: MESFUNC5:attrnot 5 => MESFUNC5:attr 5
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
  redefine attr a2 is without-infty means
    for b1 being set holds
       -infty < a2 . b1;
end;

:: MESFUNC5:dfs 5
definiens
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
To prove
     a1 is without-infty
it is sufficient to prove
  thus for b1 being set holds
       -infty < a2 . b1;

:: MESFUNC5:def 5
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
      b2 is without-infty
   iff
      for b3 being set holds
         -infty < b2 . b3;

:: MESFUNC5:attrnot 6 => MESFUNC5:attr 6
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
  redefine attr a2 is without+infty means
    for b1 being set holds
       a2 . b1 < +infty;
end;

:: MESFUNC5:dfs 6
definiens
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,ExtREAL;
To prove
     a1 is without+infty
it is sufficient to prove
  thus for b1 being set holds
       a2 . b1 < +infty;

:: MESFUNC5:def 6
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
      b2 is without+infty
   iff
      for b3 being set holds
         b2 . b3 < +infty;

:: MESFUNC5:th 16
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
      for b3 being set
            st b3 in dom b2
         holds -infty < b2 . b3
   iff
      b2 is without-infty;

:: MESFUNC5:th 17
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
      for b3 being set
            st b3 in dom b2
         holds b2 . b3 < +infty
   iff
      b2 is without+infty;

:: MESFUNC5:th 18
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
      st b2 is nonnegative
   holds b2 is without-infty;

:: MESFUNC5:th 19
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
      st b2 is nonpositive
   holds b2 is without+infty;

:: MESFUNC5:condreg 1
registration
  let a1 be non empty set;
  cluster Function-like nonnegative -> without-infty (Relation of a1,ExtREAL);
end;

:: MESFUNC5:condreg 2
registration
  let a1 be non empty set;
  cluster Function-like nonpositive -> without+infty (Relation of a1,ExtREAL);
end;

:: MESFUNC5:th 20
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,ExtREAL
      st b3 is_simple_func_in b2
   holds b3 is without+infty & b3 is without-infty;

:: MESFUNC5:th 21
theorem
for b1 being non empty set
for b2 being set
for b3 being Function-like Relation of b1,ExtREAL
      st b3 is nonnegative
   holds b3 | b2 is nonnegative;

:: MESFUNC5:th 22
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL
      st b2 is without-infty & b3 is without-infty
   holds dom (b2 + b3) = (dom b2) /\ dom b3;

:: MESFUNC5:th 23
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL
      st b2 is without-infty & b3 is without+infty
   holds dom (b2 - b3) = (dom b2) /\ dom b3;

:: MESFUNC5: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,ExtREAL
for b5 being Function-like quasi_total Relation of RAT,b2
for b6 being Element of REAL
for b7 being Element of b2
      st b3 is without-infty &
         b4 is without-infty &
         (for b8 being rational set holds
            b5 . b8 = (b7 /\ less_dom(b3,R_EAL b8)) /\ (b7 /\ less_dom(b4,R_EAL (b6 - b8))))
   holds b7 /\ less_dom(b3 + b4,R_EAL b6) = union rng b5;

:: MESFUNC5:funcnot 1 => MESFUNC5:func 1
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,REAL;
  func R_EAL A2 -> Function-like Relation of a1,ExtREAL equals
    a2;
end;

:: MESFUNC5:def 7
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
   R_EAL b2 = b2;

:: MESFUNC5: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 being sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,ExtREAL
      st b4 is nonnegative & b5 is nonnegative
   holds b4 + b5 is nonnegative;

:: MESFUNC5:th 26
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
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);

:: MESFUNC5:th 27
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL
      st for b4 being set
              st b4 in (dom b2) /\ dom b3
           holds b3 . b4 <= b2 . b4 & -infty < b3 . b4 & b2 . b4 < +infty
   holds b2 - b3 is nonnegative;

:: MESFUNC5:th 28
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL
      st b2 is nonnegative & b3 is nonnegative
   holds dom (b2 + b3) = (dom b2) /\ dom b3 &
    b2 + b3 is nonnegative;

:: MESFUNC5:th 29
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,ExtREAL
      st b2 is nonnegative & b3 is nonnegative & b4 is nonnegative
   holds dom ((b2 + b3) + b4) = ((dom b2) /\ dom b3) /\ dom b4 &
    (b2 + b3) + b4 is nonnegative &
    (for b5 being set
          st b5 in ((dom b2) /\ dom b3) /\ dom b4
       holds ((b2 + b3) + b4) . b5 = ((b2 . b5) + (b3 . b5)) + (b4 . b5));

:: MESFUNC5:th 30
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL
      st b2 is without-infty & b3 is without-infty
   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;

:: MESFUNC5:th 31
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL
      st b2 is without-infty & b2 is without+infty & b3 is without-infty & b3 is without+infty
   holds ((max+ (b2 + b3)) + max- b2) + max- b3 = ((max- (b2 + b3)) + max+ b2) + max+ b3;

:: MESFUNC5:th 32
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of REAL
      st 0 <= b3
   holds max+ (b3 (#) b2) = b3 (#) max+ b2 & max- (b3 (#) b2) = b3 (#) max- b2;

:: MESFUNC5:th 33
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of REAL
      st 0 <= b3
   holds max+ ((- b3) (#) b2) = b3 (#) max- b2 &
    max- ((- b3) (#) b2) = b3 (#) max+ b2;

:: MESFUNC5:th 34
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being set holds
   max+ (b2 | b3) = (max+ b2) | b3 & max- (b2 | b3) = (max- b2) | b3;

:: MESFUNC5:th 35
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL
for b4 being set
      st b4 c= dom (b2 + b3)
   holds dom ((b2 + b3) | b4) = b4 &
    dom ((b2 | b4) + (b3 | b4)) = b4 &
    (b2 + b3) | b4 = (b2 | b4) + (b3 | b4);

:: MESFUNC5:th 36
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of ExtREAL holds
   eq_dom(b2,b3) = b2 " {b3};

:: MESFUNC5:th 37
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,ExtREAL
for b5 being Element of b2
      st b3 is without-infty & b4 is without-infty & b3 is_measurable_on b5 & b4 is_measurable_on b5
   holds b3 + b4 is_measurable_on b5;

:: MESFUNC5:th 38
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
      st b4 is_simple_func_in b2 & dom b4 = {}
   holds ex b5 being disjoint_valued FinSequence of b2 st
      ex b6, b7 being FinSequence of ExtREAL st
         b5,b6 are_Re-presentation_of b4 &
          b6 . 1 = 0 &
          (for b8 being natural set
                st 2 <= b8 & b8 in dom b6
             holds 0 < b6 . b8 & b6 . b8 < +infty) &
          dom b7 = dom b5 &
          (for b8 being natural set
                st b8 in dom b7
             holds b7 . b8 = (b6 . b8) * ((b3 * b5) . b8)) &
          Sum b7 = 0;

:: MESFUNC5:th 39
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,ExtREAL
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,R_EAL b5)) /\ less_dom(b3,R_EAL b6) in b2;

:: MESFUNC5:th 40
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 b2
      st b4 is_simple_func_in b2
   holds b4 | b5 is_simple_func_in b2;

:: MESFUNC5:th 41
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 disjoint_valued FinSequence of b2
for b5 being Relation-like Function-like FinSequence-like set
      st dom b4 = dom b5 &
         (for b6 being natural set
               st b6 in dom b4
            holds b5 . b6 = (b4 . b6) /\ b3)
   holds b5 is disjoint_valued FinSequence of b2;

:: MESFUNC5:th 42
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,ExtREAL
for b4 being Element of b2
for b5, b6 being disjoint_valued FinSequence of b2
for b7 being FinSequence of ExtREAL
      st dom b5 = dom b6 &
         (for b8 being natural set
               st b8 in dom b5
            holds b6 . b8 = (b5 . b8) /\ b4) &
         b5,b7 are_Re-presentation_of b3
   holds b6,b7 are_Re-presentation_of b3 | b4;

:: MESFUNC5:th 43
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
      st b4 is_simple_func_in b2
   holds dom b4 is Element of b2;

:: MESFUNC5:th 44
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,ExtREAL
      st b3 is_simple_func_in b2 & b4 is_simple_func_in b2
   holds b3 + b4 is_simple_func_in b2;

:: MESFUNC5:th 45
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 b4 is_simple_func_in b2
   holds b5 (#) b4 is_simple_func_in b2;

:: MESFUNC5: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 sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,ExtREAL
      st b4 is_simple_func_in b2 &
         b5 is_simple_func_in b2 &
         (for b6 being set
               st b6 in dom (b4 - b5)
            holds b5 . b6 <= b4 . b6)
   holds b4 - b5 is nonnegative;

:: MESFUNC5: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 sigma_Measure of b2
for b4 being Element of b2
for b5 being Element of ExtREAL
      st b5 <> +infty & b5 <> -infty
   holds ex b6 being Function-like Relation of b1,ExtREAL st
      b6 is_simple_func_in b2 &
       dom b6 = b4 &
       (for b7 being set
             st b7 in b4
          holds b6 . b7 = b5);

:: MESFUNC5: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 sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL
for b5, b6 being Element of b2
      st b4 is_measurable_on b5 & b6 = (dom b4) /\ b5
   holds b4 | b5 is_measurable_on b6;

:: MESFUNC5: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 sigma_Measure of b2
for b4 being Element of b2
for b5, b6 being Function-like Relation of b1,ExtREAL
      st b4 c= dom b5 & b5 is_measurable_on b4 & b6 is_measurable_on b4 & b5 is without-infty & b6 is without-infty
   holds (max+ (b5 + b6)) + max- b5 is_measurable_on b4;

:: MESFUNC5: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 sigma_Measure of b2
for b4 being Element of b2
for b5, b6 being Function-like Relation of b1,ExtREAL
      st b4 c= (dom b5) /\ dom b6 & b5 is_measurable_on b4 & b6 is_measurable_on b4 & b5 is without-infty & b6 is without-infty
   holds (max- (b5 + b6)) + max+ b5 is_measurable_on b4;

:: MESFUNC5:th 51
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 set
      st b4 in b2
   holds 0 <= b3 . b4;

:: MESFUNC5:th 52
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,ExtREAL
      st (ex b6 being Element of b2 st
            b6 = dom b4 & b4 is_measurable_on b6) &
         (ex b6 being Element of b2 st
            b6 = dom b5 & b5 is_measurable_on b6) &
         b4 " {+infty} in b2 &
         b4 " {-infty} in b2 &
         b5 " {+infty} in b2 &
         b5 " {-infty} in b2
   holds dom (b4 + b5) in b2;

:: MESFUNC5:th 53
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,ExtREAL
      st (ex b6 being Element of b2 st
            b6 = dom b4 & b4 is_measurable_on b6) &
         (ex b6 being Element of b2 st
            b6 = dom b5 & b5 is_measurable_on b6)
   holds ex b6 being Element of b2 st
      b6 = dom (b4 + b5) & b4 + b5 is_measurable_on b6;

:: MESFUNC5:th 54
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, b6 being Element of b2
      st dom b4 = b5
   holds    b4 is_measurable_on b6
   iff
      b4 is_measurable_on b5 /\ b6;

:: MESFUNC5:th 55
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
   st ex b5 being Element of b2 st
        dom b4 = b5
for b5 being Element of REAL
for b6 being Element of b2
      st b4 is_measurable_on b6
   holds b5 (#) b4 is_measurable_on b6;

:: MESFUNC5:modenot 1
definition
  mode ExtREAL_sequence is Function-like quasi_total Relation of NAT,ExtREAL;
end;

:: MESFUNC5:attrnot 7 => MESFUNC5:attr 7
definition
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
  attr a1 is convergent_to_finite_number means
    ex b1 being real set st
       for b2 being real set
             st 0 < b2
          holds ex b3 being natural set st
             for b4 being natural set
                   st b3 <= b4
                holds |.(a1 . b4) - R_EAL b1.| < b2;
end;

:: MESFUNC5:dfs 8
definiens
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
To prove
     a1 is convergent_to_finite_number
it is sufficient to prove
  thus ex b1 being real set st
       for b2 being real set
             st 0 < b2
          holds ex b3 being natural set st
             for b4 being natural set
                   st b3 <= b4
                holds |.(a1 . b4) - R_EAL b1.| < b2;

:: MESFUNC5:def 8
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
      b1 is convergent_to_finite_number
   iff
      ex b2 being real set st
         for b3 being real set
               st 0 < b3
            holds ex b4 being natural set st
               for b5 being natural set
                     st b4 <= b5
                  holds |.(b1 . b5) - R_EAL b2.| < b3;

:: MESFUNC5:attrnot 8 => MESFUNC5:attr 8
definition
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
  attr a1 is convergent_to_+infty means
    for b1 being real set
          st 0 < b1
       holds ex b2 being natural set st
          for b3 being natural set
                st b2 <= b3
             holds b1 <= a1 . b3;
end;

:: MESFUNC5:dfs 9
definiens
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
To prove
     a1 is convergent_to_+infty
it is sufficient to prove
  thus for b1 being real set
          st 0 < b1
       holds ex b2 being natural set st
          for b3 being natural set
                st b2 <= b3
             holds b1 <= a1 . b3;

:: MESFUNC5:def 9
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
      b1 is convergent_to_+infty
   iff
      for b2 being real set
            st 0 < b2
         holds ex b3 being natural set st
            for b4 being natural set
                  st b3 <= b4
               holds b2 <= b1 . b4;

:: MESFUNC5:attrnot 9 => MESFUNC5:attr 9
definition
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
  attr a1 is convergent_to_-infty means
    for b1 being real set
          st b1 < 0
       holds ex b2 being natural set st
          for b3 being natural set
                st b2 <= b3
             holds a1 . b3 <= b1;
end;

:: MESFUNC5:dfs 10
definiens
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
To prove
     a1 is convergent_to_-infty
it is sufficient to prove
  thus for b1 being real set
          st b1 < 0
       holds ex b2 being natural set st
          for b3 being natural set
                st b2 <= b3
             holds a1 . b3 <= b1;

:: MESFUNC5:def 10
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
      b1 is convergent_to_-infty
   iff
      for b2 being real set
            st b2 < 0
         holds ex b3 being natural set st
            for b4 being natural set
                  st b3 <= b4
               holds b1 . b4 <= b2;

:: MESFUNC5:th 56
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
      st b1 is convergent_to_+infty
   holds b1 is not convergent_to_-infty & b1 is not convergent_to_finite_number;

:: MESFUNC5:th 57
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
      st b1 is convergent_to_-infty
   holds b1 is not convergent_to_+infty & b1 is not convergent_to_finite_number;

:: MESFUNC5:attrnot 10 => MESFUNC5:attr 10
definition
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
  attr a1 is convergent means
    (a1 is not convergent_to_finite_number & a1 is not convergent_to_+infty) implies a1 is convergent_to_-infty;
end;

:: MESFUNC5:dfs 11
definiens
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
To prove
     a1 is convergent
it is sufficient to prove
  thus (a1 is not convergent_to_finite_number & a1 is not convergent_to_+infty) implies a1 is convergent_to_-infty;

:: MESFUNC5:def 11
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
      b1 is convergent
   iff
      (b1 is not convergent_to_finite_number & b1 is not convergent_to_+infty implies b1 is convergent_to_-infty);

:: MESFUNC5:funcnot 2 => MESFUNC5:func 2
definition
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
  assume a1 is convergent;
  func lim A1 -> Element of ExtREAL means
    ((for b1 being real set
           st it = b1 &
              (for b2 being real set
                    st 0 < b2
                 holds ex b3 being natural set st
                    for b4 being natural set
                          st b3 <= b4
                       holds |.(a1 . b4) - it.| < b2)
        holds a1 is not convergent_to_finite_number) &
     (it = +infty implies a1 is not convergent_to_+infty)) implies it = -infty & a1 is convergent_to_-infty;
end;

:: MESFUNC5:def 12
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
   st b1 is convergent
for b2 being Element of ExtREAL holds
      b2 = lim b1
   iff
      ((for b3 being real set
             st b2 = b3 &
                (for b4 being real set
                      st 0 < b4
                   holds ex b5 being natural set st
                      for b6 being natural set
                            st b5 <= b6
                         holds |.(b1 . b6) - b2.| < b4)
          holds b1 is not convergent_to_finite_number) &
       (b2 = +infty implies b1 is not convergent_to_+infty) implies b2 = -infty & b1 is convergent_to_-infty);

:: MESFUNC5:th 58
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being real set
      st for b3 being natural set holds
           b1 . b3 = b2
   holds b1 is convergent_to_finite_number & lim b1 = b2;

:: MESFUNC5:th 59
theorem
for b1 being FinSequence of ExtREAL
      st for b2 being natural set
              st b2 in dom b1
           holds 0 <= b1 . b2
   holds 0 <= Sum b1;

:: MESFUNC5:th 60
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
      st for b2, b3 being natural set
              st b2 <= b3
           holds b1 . b2 <= b1 . b3
   holds b1 is convergent & lim b1 = sup rng b1;

:: MESFUNC5:th 61
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
      st for b3 being natural set holds
           b1 . b3 <= b2 . b3
   holds sup rng b1 <= sup rng b2;

:: MESFUNC5:th 62
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being natural set holds
   b1 . b2 <= sup rng b1;

:: MESFUNC5:th 63
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Element of ExtREAL
      st for b3 being natural set holds
           b1 . b3 <= b2
   holds sup rng b1 <= b2;

:: MESFUNC5:th 64
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Element of ExtREAL
      st b2 <> +infty &
         (for b3 being natural set holds
            b1 . b3 <= b2)
   holds sup rng b1 < +infty;

:: MESFUNC5:th 65
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
      st b1 is without-infty
   holds    sup rng b1 <> +infty
   iff
      ex b2 being real set st
         0 < b2 &
          (for b3 being natural set holds
             b1 . b3 <= b2);

:: MESFUNC5:th 66
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Element of ExtREAL
      st for b3 being natural set holds
           b1 . b3 = b2
   holds b1 is convergent & lim b1 = b2 & lim b1 = sup rng b1;

:: MESFUNC5:th 67
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,ExtREAL
      st (for b4, b5 being natural set
               st b4 <= b5
            holds b1 . b4 <= b1 . b5) &
         (for b4, b5 being natural set
               st b4 <= b5
            holds b2 . b4 <= b2 . b5) &
         b1 is without-infty &
         b2 is without-infty &
         (for b4 being natural set holds
            (b1 . b4) + (b2 . b4) = b3 . b4)
   holds b3 is convergent &
    lim b3 = sup rng b3 &
    lim b3 = (lim b1) + lim b2 &
    sup rng b3 = (sup rng b2) + sup rng b1;

:: MESFUNC5:th 68
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
for b3 being Element of REAL
      st 0 <= b3 &
         b1 is without-infty &
         (for b4 being natural set holds
            b2 . b4 = (R_EAL b3) * (b1 . b4))
   holds sup rng b2 = (R_EAL b3) * sup rng b1 &
    b2 is without-infty;

:: MESFUNC5:th 69
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
for b3 being Element of REAL
      st 0 <= b3 &
         (for b4, b5 being natural set
               st b4 <= b5
            holds b1 . b4 <= b1 . b5) &
         (for b4 being natural set holds
            b2 . b4 = (R_EAL b3) * (b1 . b4)) &
         b1 is without-infty
   holds (for b4, b5 being natural set
          st b4 <= b5
       holds b2 . b4 <= b2 . b5) &
    b2 is without-infty &
    b2 is convergent &
    lim b2 = sup rng b2 &
    lim b2 = (R_EAL b3) * lim b1;

:: MESFUNC5:funcnot 3 => MESFUNC5:func 3
definition
  let a1 be non empty set;
  let a2 be Functional_Sequence of a1,ExtREAL;
  let a3 be Element of a1;
  func A2 # A3 -> Function-like quasi_total Relation of NAT,ExtREAL means
    for b1 being natural set holds
       it . b1 = (a2 . b1) . a3;
end;

:: MESFUNC5:def 13
theorem
for b1 being non empty set
for b2 being Functional_Sequence of b1,ExtREAL
for b3 being Element of b1
for b4 being Function-like quasi_total Relation of NAT,ExtREAL holds
      b4 = b2 # b3
   iff
      for b5 being natural set holds
         b4 . b5 = (b2 . b5) . b3;

:: MESFUNC5:funcnot 4 => MESFUNC5:func 4
definition
  let a1, a2 be set;
  let a3 be Function-like quasi_total Relation of NAT,PFuncs(a1,a2);
  let a4 be natural set;
  redefine func a3 . a4 -> Function-like Relation of a1,a2;
end;

:: MESFUNC5: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,ExtREAL
      st (ex b4 being Element of b2 st
            b4 = dom b3 & b3 is_measurable_on b4) &
         b3 is nonnegative
   holds ex b4 being Functional_Sequence of b1,ExtREAL st
      (for b5 being natural set holds
          b4 . b5 is_simple_func_in b2 & dom (b4 . b5) = dom b3) &
       (for b5 being natural set holds
          b4 . b5 is nonnegative) &
       (for b5, b6 being natural set
          st b5 <= b6
       for b7 being Element of b1
             st b7 in dom b3
          holds (b4 . b5) . b7 <= (b4 . b6) . b7) &
       (for b5 being Element of b1
             st b5 in dom b3
          holds b4 # b5 is convergent & lim (b4 # b5) = b3 . b5);

:: MESFUNC5:funcnot 5 => MESFUNC5:func 5
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,ExtREAL;
  func integral'(A3,A4) -> Element of ExtREAL equals
    integral(a1,a2,a3,a4)
    if dom a4 <> {}
    otherwise 0.;
end;

:: MESFUNC5:def 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 sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL holds
   (dom b4 = {} or integral'(b3,b4) = integral(b1,b2,b3,b4)) &
    (dom b4 = {} implies integral'(b3,b4) = 0.);

:: MESFUNC5: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 sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,ExtREAL
      st b4 is_simple_func_in b2 & b5 is_simple_func_in b2 & b4 is nonnegative & b5 is nonnegative
   holds dom (b4 + b5) = (dom b4) /\ dom b5 &
    integral'(b3,b4 + b5) = (integral'(b3,b4 | dom (b4 + b5))) + integral'(b3,b5 | dom (b4 + b5));

:: MESFUNC5: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 being sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL
for b5 being Element of REAL
      st b4 is_simple_func_in b2 & b4 is nonnegative & 0 <= b5
   holds integral'(b3,b5 (#) b4) = (R_EAL b5) * integral'(b3,b4);

:: MESFUNC5: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 sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL
for b5, b6 being Element of b2
      st b4 is_simple_func_in b2 & b4 is nonnegative & b5 misses b6
   holds integral'(b3,b4 | (b5 \/ b6)) = (integral'(b3,b4 | b5)) + integral'(b3,b4 | b6);

:: MESFUNC5:th 74
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
      st b4 is_simple_func_in b2 & b4 is nonnegative
   holds 0 <= integral'(b3,b4);

:: MESFUNC5: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 sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,ExtREAL
      st b4 is_simple_func_in b2 &
         b4 is nonnegative &
         b5 is_simple_func_in b2 &
         b5 is nonnegative &
         (for b6 being set
               st b6 in dom (b4 - b5)
            holds b5 . b6 <= b4 . b6)
   holds dom (b4 - b5) = (dom b4) /\ dom b5 &
    integral'(b3,b4 | dom (b4 - b5)) = (integral'(b3,b4 - b5)) + integral'(b3,b5 | dom (b4 - b5));

:: MESFUNC5: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 sigma_Measure of b2
for b4, b5 being Function-like Relation of b1,ExtREAL
      st b4 is_simple_func_in b2 &
         b5 is_simple_func_in b2 &
         b4 is nonnegative &
         b5 is nonnegative &
         (for b6 being set
               st b6 in dom (b4 - b5)
            holds b5 . b6 <= b4 . b6)
   holds integral'(b3,b5 | dom (b4 - b5)) <= integral'(b3,b4 | dom (b4 - b5));

:: MESFUNC5: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 being sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL
for b5 being Element of ExtREAL
      st 0 <= b5 &
         b4 is_simple_func_in b2 &
         (for b6 being set
               st b6 in dom b4
            holds b4 . b6 = b5)
   holds integral'(b3,b4) = b5 * (b3 . dom b4);

:: MESFUNC5: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 being sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL
      st b4 is_simple_func_in b2 & b4 is nonnegative
   holds integral'(b3,b4 | eq_dom(b4,R_EAL 0)) = 0;

:: MESFUNC5: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 being sigma_Measure of b2
for b4 being Element of b2
for b5 being Function-like Relation of b1,ExtREAL
      st b5 is_simple_func_in b2 & b3 . b4 = 0 & b5 is nonnegative
   holds integral'(b3,b5 | b4) = 0;

:: MESFUNC5: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 sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL
for b5 being Functional_Sequence of b1,ExtREAL
for b6 being Function-like quasi_total Relation of NAT,ExtREAL
      st b4 is_simple_func_in b2 &
         (for b7 being set
               st b7 in dom b4
            holds 0 < b4 . b7) &
         (for b7 being natural set holds
            b5 . b7 is_simple_func_in b2) &
         (for b7 being natural set holds
            dom (b5 . b7) = dom b4) &
         (for b7 being natural set holds
            b5 . b7 is nonnegative) &
         (for b7, b8 being natural set
            st b7 <= b8
         for b9 being Element of b1
               st b9 in dom b4
            holds (b5 . b7) . b9 <= (b5 . b8) . b9) &
         (for b7 being Element of b1
               st b7 in dom b4
            holds b5 # b7 is convergent & b4 . b7 <= lim (b5 # b7)) &
         (for b7 being natural set holds
            b6 . b7 = integral'(b3,b5 . b7))
   holds b6 is convergent & integral'(b3,b4) <= lim b6;

:: MESFUNC5: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 sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL
for b5 being Functional_Sequence of b1,ExtREAL
      st b4 is_simple_func_in b2 &
         b4 is nonnegative &
         (for b6 being natural set holds
            b5 . b6 is_simple_func_in b2) &
         (for b6 being natural set holds
            dom (b5 . b6) = dom b4) &
         (for b6 being natural set holds
            b5 . b6 is nonnegative) &
         (for b6, b7 being natural set
            st b6 <= b7
         for b8 being Element of b1
               st b8 in dom b4
            holds (b5 . b6) . b8 <= (b5 . b7) . b8) &
         (for b6 being Element of b1
               st b6 in dom b4
            holds b5 # b6 is convergent & b4 . b6 <= lim (b5 # b6))
   holds ex b6 being Function-like quasi_total Relation of NAT,ExtREAL st
      (for b7 being natural set holds
          b6 . b7 = integral'(b3,b5 . b7)) &
       b6 is convergent &
       sup rng b6 = lim b6 &
       integral'(b3,b4) <= lim b6;

:: MESFUNC5: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 Element of b2
for b5, b6 being Functional_Sequence of b1,ExtREAL
for b7, b8 being Function-like quasi_total Relation of NAT,ExtREAL
      st (for b9 being natural set holds
            b5 . b9 is_simple_func_in b2 & dom (b5 . b9) = b4) &
         (for b9 being natural set holds
            b5 . b9 is nonnegative) &
         (for b9, b10 being natural set
            st b9 <= b10
         for b11 being Element of b1
               st b11 in b4
            holds (b5 . b9) . b11 <= (b5 . b10) . b11) &
         (for b9 being natural set holds
            b6 . b9 is_simple_func_in b2 & dom (b6 . b9) = b4) &
         (for b9 being natural set holds
            b6 . b9 is nonnegative) &
         (for b9, b10 being natural set
            st b9 <= b10
         for b11 being Element of b1
               st b11 in b4
            holds (b6 . b9) . b11 <= (b6 . b10) . b11) &
         (for b9 being Element of b1
               st b9 in b4
            holds b5 # b9 is convergent & b6 # b9 is convergent & lim (b5 # b9) = lim (b6 # b9)) &
         (for b9 being natural set holds
            b7 . b9 = integral'(b3,b5 . b9) & b8 . b9 = integral'(b3,b6 . b9))
   holds b7 is convergent & b8 is convergent & lim b7 = lim b8;

:: MESFUNC5:funcnot 6 => MESFUNC5:func 6
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,ExtREAL;
  assume (ex b1 being Element of a2 st
        b1 = dom a4 & a4 is_measurable_on b1) &
     a4 is nonnegative;
  func integral+(A3,A4) -> Element of ExtREAL means
    ex b1 being Functional_Sequence of a1,ExtREAL st
       ex b2 being Function-like quasi_total Relation of NAT,ExtREAL st
          (for b3 being natural set holds
              b1 . b3 is_simple_func_in a2 & dom (b1 . b3) = dom a4) &
           (for b3 being natural set holds
              b1 . b3 is nonnegative) &
           (for b3, b4 being natural set
              st b3 <= b4
           for b5 being Element of a1
                 st b5 in dom a4
              holds (b1 . b3) . b5 <= (b1 . b4) . b5) &
           (for b3 being Element of a1
                 st b3 in dom a4
              holds b1 # b3 is convergent & lim (b1 # b3) = a4 . b3) &
           (for b3 being natural set holds
              b2 . b3 = integral'(a3,b1 . b3)) &
           b2 is convergent &
           it = lim b2;
end;

:: MESFUNC5:def 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 sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL
   st (ex b5 being Element of b2 st
         b5 = dom b4 & b4 is_measurable_on b5) &
      b4 is nonnegative
for b5 being Element of ExtREAL holds
      b5 = integral+(b3,b4)
   iff
      ex b6 being Functional_Sequence of b1,ExtREAL st
         ex b7 being Function-like quasi_total Relation of NAT,ExtREAL st
            (for b8 being natural set holds
                b6 . b8 is_simple_func_in b2 & dom (b6 . b8) = dom b4) &
             (for b8 being natural set holds
                b6 . b8 is nonnegative) &
             (for b8, b9 being natural set
                st b8 <= b9
             for b10 being Element of b1
                   st b10 in dom b4
                holds (b6 . b8) . b10 <= (b6 . b9) . b10) &
             (for b8 being Element of b1
                   st b8 in dom b4
                holds b6 # b8 is convergent & lim (b6 # b8) = b4 . b8) &
             (for b8 being natural set holds
                b7 . b8 = integral'(b3,b6 . b8)) &
             b7 is convergent &
             b5 = lim b7;

:: MESFUNC5: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,ExtREAL
      st b4 is_simple_func_in b2 & b4 is nonnegative
   holds integral+(b3,b4) = integral'(b3,b4);

:: MESFUNC5: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, b5 being Function-like Relation of b1,ExtREAL
      st (ex b6 being Element of b2 st
            b6 = dom b4 & b4 is_measurable_on b6) &
         (ex b6 being Element of b2 st
            b6 = dom b5 & b5 is_measurable_on b6) &
         b4 is nonnegative &
         b5 is nonnegative
   holds ex b6 being Element of b2 st
      b6 = dom (b4 + b5) &
       integral+(b3,b4 + b5) = (integral+(b3,b4 | b6)) + integral+(b3,b5 | b6);

:: MESFUNC5: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,ExtREAL
      st (ex b5 being Element of b2 st
            b5 = dom b4 & b4 is_measurable_on b5) &
         b4 is nonnegative
   holds 0 <= integral+(b3,b4);

:: MESFUNC5: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,ExtREAL
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);

:: MESFUNC5: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,ExtREAL
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);

:: MESFUNC5: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,ExtREAL
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 &
         b3 . b5 = 0
   holds integral+(b3,b4 | b5) = 0;

:: MESFUNC5: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,ExtREAL
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);

:: MESFUNC5: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,ExtREAL
for b5, b6 being Element of b2
      st b4 is nonnegative & b5 = dom b4 & b4 is_measurable_on b5 & b3 . b6 = 0
   holds integral+(b3,b4 | (b5 \ b6)) = integral+(b3,b4);

:: MESFUNC5: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, b5 being Function-like Relation of b1,ExtREAL
      st (ex b6 being Element of b2 st
            b6 = dom b4 & b6 = dom b5 & b4 is_measurable_on b6 & b5 is_measurable_on b6) &
         b4 is nonnegative &
         b5 is nonnegative &
         (for b6 being Element of b1
               st b6 in dom b5
            holds b5 . b6 <= b4 . b6)
   holds integral+(b3,b5) <= integral+(b3,b4);

:: MESFUNC5: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,ExtREAL
for b5 being Element of REAL
      st 0 <= b5 &
         (ex b6 being Element of b2 st
            b6 = dom b4 & b4 is_measurable_on b6) &
         b4 is nonnegative
   holds integral+(b3,b5 (#) b4) = (R_EAL b5) * integral+(b3,b4);

:: MESFUNC5: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,ExtREAL
      st (ex b5 being Element of b2 st
            b5 = dom b4 & b4 is_measurable_on b5) &
         (for b5 being Element of b1
               st b5 in dom b4
            holds 0 = b4 . b5)
   holds integral+(b3,b4) = 0;

:: MESFUNC5:funcnot 7 => MESFUNC5:func 7
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,ExtREAL;
  func Integral(A3,A4) -> Element of ExtREAL equals
    (integral+(a3,max+ a4)) - integral+(a3,max- a4);
end;

:: MESFUNC5:def 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 sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL holds
   Integral(b3,b4) = (integral+(b3,max+ b4)) - integral+(b3,max- b4);

:: MESFUNC5: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,ExtREAL
      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,b4);

:: MESFUNC5: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,ExtREAL
      st b4 is_simple_func_in b2 & b4 is nonnegative
   holds Integral(b3,b4) = integral+(b3,b4) & Integral(b3,b4) = integral'(b3,b4);

:: MESFUNC5: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 being Function-like Relation of b1,ExtREAL
      st (ex b5 being Element of b2 st
            b5 = dom b4 & b4 is_measurable_on b5) &
         b4 is nonnegative
   holds 0 <= Integral(b3,b4);

:: MESFUNC5: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,ExtREAL
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);

:: MESFUNC5: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 being Function-like Relation of b1,ExtREAL
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);

:: MESFUNC5: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 being Function-like Relation of b1,ExtREAL
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);

:: MESFUNC5: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 being Function-like Relation of b1,ExtREAL
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;

:: MESFUNC5: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 being Function-like Relation of b1,ExtREAL
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);

:: MESFUNC5:prednot 1 => MESFUNC5: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 sigma_Measure of a2;
  let a4 be Function-like Relation of a1,ExtREAL;
  pred A4 is_integrable_on A3 means
    (ex b1 being Element of a2 st
        b1 = dom a4 & a4 is_measurable_on b1) &
     integral+(a3,max+ a4) < +infty &
     integral+(a3,max- a4) < +infty;
end;

:: MESFUNC5:dfs 17
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,ExtREAL;
To prove
     a4 is_integrable_on a3
it is sufficient to prove
  thus (ex b1 being Element of a2 st
        b1 = dom a4 & a4 is_measurable_on b1) &
     integral+(a3,max+ a4) < +infty &
     integral+(a3,max- a4) < +infty;

:: MESFUNC5:def 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 sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL holds
      b4 is_integrable_on b3
   iff
      (ex b5 being Element of b2 st
          b5 = dom b4 & b4 is_measurable_on b5) &
       integral+(b3,max+ b4) < +infty &
       integral+(b3,max- b4) < +infty;

:: MESFUNC5: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,ExtREAL
      st b4 is_integrable_on b3
   holds 0 <= integral+(b3,max+ b4) & 0 <= integral+(b3,max- b4) & -infty < Integral(b3,b4) & Integral(b3,b4) < +infty;

:: MESFUNC5: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 being Function-like Relation of b1,ExtREAL
for b5 being Element of b2
      st b4 is_integrable_on b3
   holds integral+(b3,max+ (b4 | b5)) <= integral+(b3,max+ b4) &
    integral+(b3,max- (b4 | b5)) <= integral+(b3,max- b4) &
    b4 | b5 is_integrable_on b3;

:: MESFUNC5: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,ExtREAL
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);

:: MESFUNC5:th 105
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, b6 being Element of b2
      st b4 is_integrable_on b3 & b6 = (dom b4) \ b5
   holds b4 | b5 is_integrable_on b3 &
    Integral(b3,b4) = (Integral(b3,b4 | b5)) + Integral(b3,b4 | b6);

:: MESFUNC5:th 106
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
      st ex b5 being Element of b2 st
           b5 = dom b4 & b4 is_measurable_on b5
   holds    b4 is_integrable_on b3
   iff
      |.b4.| is_integrable_on b3;

:: MESFUNC5:th 107
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
      st b4 is_integrable_on b3
   holds |.Integral(b3,b4).| <= Integral(b3,|.b4.|);

:: MESFUNC5:th 108
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,ExtREAL
      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 |.b4 . b6.| <= b5 . b6)
   holds b4 is_integrable_on b3 &
    Integral(b3,|.b4.|) <= Integral(b3,b5);

:: MESFUNC5:th 109
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 &
         0 <= b5 &
         dom b4 <> {} &
         (for b6 being set
               st b6 in dom b4
            holds b4 . b6 = b5)
   holds integral(b1,b2,b3,b4) = (R_EAL b5) * (b3 . dom b4);

:: MESFUNC5:th 110
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 &
         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);

:: MESFUNC5:th 111
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
      st b4 is_integrable_on b3
   holds b4 " {+infty} in b2 &
    b4 " {-infty} in b2 &
    b3 . (b4 " {+infty}) = 0 &
    b3 . (b4 " {-infty}) = 0 &
    (b4 " {+infty}) \/ (b4 " {-infty}) in b2 &
    b3 . ((b4 " {+infty}) \/ (b4 " {-infty})) = 0;

:: MESFUNC5:th 112
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,ExtREAL
      st b4 is_integrable_on b3 & b5 is_integrable_on b3 & b4 is nonnegative & b5 is nonnegative
   holds b4 + b5 is_integrable_on b3;

:: MESFUNC5:th 113
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,ExtREAL
      st b4 is_integrable_on b3 & b5 is_integrable_on b3
   holds dom (b4 + b5) in b2;

:: MESFUNC5:th 114
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,ExtREAL
      st b4 is_integrable_on b3 & b5 is_integrable_on b3
   holds b4 + b5 is_integrable_on b3;

:: MESFUNC5:th 115
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,ExtREAL
      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);

:: MESFUNC5:th 116
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 b4 is_integrable_on b3
   holds b5 (#) b4 is_integrable_on b3 &
    Integral(b3,b5 (#) b4) = (R_EAL b5) * Integral(b3,b4);

:: MESFUNC5:funcnot 8 => MESFUNC5:func 8
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,ExtREAL;
  let a5 be Element of a2;
  func Integral_on(A3,A5,A4) -> Element of ExtREAL equals
    Integral(a3,a4 | a5);
end;

:: MESFUNC5:def 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 sigma_Measure of b2
for b4 being Function-like Relation of b1,ExtREAL
for b5 being Element of b2 holds
   Integral_on(b3,b5,b4) = Integral(b3,b4 | b5);

:: MESFUNC5:th 117
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,ExtREAL
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);

:: MESFUNC5:th 118
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
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);