:: MESFUNC5 semantic presentation

theorem Th1: :: MESFUNC5:1
for b1, b2 being R_eal holds |.(b1 - b2).| = |.(b2 - b1).|
proof end;

theorem Th2: :: MESFUNC5:2
for b1, b2 being R_eal holds b2 - b1 <= |.(b1 - b2).|
proof end;

theorem Th3: :: MESFUNC5:3
for b1, b2 being R_eal
for b3 being real number holds
not ( |.(b1 - b2).| < b3 & not ( b1 = +infty & b2 = +infty ) & not ( b1 = -infty & b2 = -infty ) & not ( b1 <> +infty & b1 <> -infty & b2 <> +infty & b2 <> -infty ) )
proof end;

theorem Th4: :: MESFUNC5:4
for b1, b2 being R_eal holds
( ( for b3 being real number holds
not ( 0 < b3 & not b1 < b2 + (R_EAL b3) ) ) implies b1 <= b2 )
proof end;

theorem Th5: :: MESFUNC5:5
for b1, b2, b3 being R_eal holds
not ( b3 <> -infty & b3 <> +infty & b1 < b2 & not b1 + b3 < b2 + b3 )
proof end;

theorem Th6: :: MESFUNC5:6
for b1, b2, b3 being R_eal holds
not ( b3 <> -infty & b3 <> +infty & b1 < b2 & not b1 - b3 < b2 - b3 )
proof end;

theorem Th7: :: MESFUNC5:7
for b1, b2 being real number holds
( (R_EAL b1) + (R_EAL b2) = b1 + b2 & - (R_EAL b1) = - b1 )
proof end;

theorem Th8: :: MESFUNC5:8
for b1 being Nat
for b2 being R_eal holds
not ( 0 <= b2 & b2 < b1 & ( for b3 being Nat holds
not ( 1 <= b3 & b3 <= (2 |^ b1) * b1 & (b3 - 1) / (2 |^ b1) <= b2 & b2 < b3 / (2 |^ b1) ) ) )
proof end;

theorem Th9: :: MESFUNC5:9
for b1, b2 being Nat
for b3 being R_eal holds
( 1 <= b2 & b2 <= (2 |^ b1) * b1 & b1 <= b3 & (b2 - 1) / (2 |^ b1) <= b3 implies b2 / (2 |^ b1) <= b3 )
proof end;

theorem Th10: :: MESFUNC5:10
for b1, b2, b3, b4 being R_eal holds
not ( -infty < b3 & b1 < b2 & b3 < b4 & not b1 + b3 < b2 + b4 )
proof end;

theorem Th11: :: MESFUNC5:11
for b1, b2, b3 being R_eal holds
( 0 <= b3 implies ( b3 * (max b1,b2) = max (b3 * b1),(b3 * b2) & b3 * (min b1,b2) = min (b3 * b1),(b3 * b2) ) )
proof end;

theorem Th12: :: MESFUNC5:12
for b1, b2, b3 being R_eal holds
( b3 <= 0 implies ( b3 * (min b1,b2) = max (b3 * b1),(b3 * b2) & b3 * (max b1,b2) = min (b3 * b1),(b3 * b2) ) )
proof end;

theorem Th13: :: MESFUNC5:13
for b1, b2, b3 being R_eal holds
( 0 <= b1 & 0 <= b3 & b3 + b1 <= b2 implies b3 <= b2 )
proof end;

definition
let c1 be set ;
attr a1 is nonpositive means :Def1: :: MESFUNC5:def 1
for b1 being R_eal holds
( b1 in a1 implies b1 <= 0 );
end;

:: deftheorem Def1 defines nonpositive MESFUNC5:def 1 :
for b1 being set holds
( b1 is nonpositive iff for b2 being R_eal holds
( b2 in b1 implies b2 <= 0 ) );

definition
let c1 be Relation;
attr a1 is nonpositive means :Def2: :: MESFUNC5:def 2
rng a1 is nonpositive;
end;

:: deftheorem Def2 defines nonpositive MESFUNC5:def 2 :
for b1 being Relation holds
( b1 is nonpositive iff rng b1 is nonpositive );

theorem Th14: :: MESFUNC5:14
for b1 being set
for b2 being PartFunc of b1, ExtREAL holds
( b2 is nonpositive iff for b3 being set holds b2 . b3 <= 0. )
proof end;

theorem Th15: :: MESFUNC5:15
for b1 being set
for b2 being PartFunc of b1, ExtREAL holds
( ( for b3 being set holds
( b3 in dom b2 implies b2 . b3 <= 0. ) ) implies b2 is nonpositive )
proof end;

definition
let c1 be Relation;
attr a1 is without-infty means :Def3: :: MESFUNC5:def 3
not -infty in rng a1;
attr a1 is without+infty means :Def4: :: MESFUNC5:def 4
not +infty in rng a1;
end;

:: deftheorem Def3 defines without-infty MESFUNC5:def 3 :
for b1 being Relation holds
( b1 is without-infty iff not -infty in rng b1 );

:: deftheorem Def4 defines without+infty MESFUNC5:def 4 :
for b1 being Relation holds
( b1 is without+infty iff not +infty in rng b1 );

definition
let c1 be non empty set ;
let c2 be PartFunc of c1, ExtREAL ;
redefine attr without-infty as a2 is without-infty means :Def5: :: MESFUNC5:def 5
for b1 being set holds
-infty < a2 . b1;
compatibility
( c2 is without-infty iff for b1 being set holds
-infty < c2 . b1 )
proof end;
redefine attr without+infty as a2 is without+infty means :Def6: :: MESFUNC5:def 6
for b1 being set holds
a2 . b1 < +infty ;
compatibility
( c2 is without+infty iff for b1 being set holds
c2 . b1 < +infty )
proof end;
end;

:: deftheorem Def5 defines without-infty MESFUNC5:def 5 :
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL holds
( b2 is without-infty iff for b3 being set holds
-infty < b2 . b3 );

:: deftheorem Def6 defines without+infty MESFUNC5:def 6 :
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL holds
( b2 is without+infty iff for b3 being set holds
b2 . b3 < +infty );

theorem Th16: :: MESFUNC5:16
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL holds
( ( for b3 being set holds
not ( b3 in dom b2 & not -infty < b2 . b3 ) ) iff b2 is without-infty )
proof end;

theorem Th17: :: MESFUNC5:17
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL holds
( ( for b3 being set holds
not ( b3 in dom b2 & not b2 . b3 < +infty ) ) iff b2 is without+infty )
proof end;

theorem Th18: :: MESFUNC5:18
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL holds
( b2 is nonnegative implies b2 is without-infty )
proof end;

theorem Th19: :: MESFUNC5:19
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL holds
( b2 is nonpositive implies b2 is without+infty )
proof end;

registration
let c1 be non empty set ;
cluster nonnegative -> V174 Relation of a1, ExtREAL ;
coherence
for b1 being PartFunc of c1, ExtREAL holds
( b1 is nonnegative implies b1 is without-infty )
by Th18;
cluster nonpositive -> V175 Relation of a1, ExtREAL ;
coherence
for b1 being PartFunc of c1, ExtREAL holds
( b1 is nonpositive implies b1 is without+infty )
by Th19;
end;

theorem Th20: :: MESFUNC5:20
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL holds
( b3 is_simple_func_in b2 implies ( b3 is without+infty & b3 is without-infty ) )
proof end;

theorem Th21: :: MESFUNC5:21
for b1 being non empty set
for b2 being set
for b3 being PartFunc of b1, ExtREAL holds
( b3 is nonnegative implies b3 | b2 is nonnegative )
proof end;

theorem Th22: :: MESFUNC5:22
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL holds
( b2 is without-infty & b3 is without-infty implies dom (b2 + b3) = (dom b2) /\ (dom b3) )
proof end;

theorem Th23: :: MESFUNC5:23
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL holds
( b2 is without-infty & b3 is without+infty implies dom (b2 - b3) = (dom b2) /\ (dom b3) )
proof end;

theorem Th24: :: MESFUNC5:24
for b1 being non empty set
for b2 being SigmaField of b1
for b3, b4 being PartFunc of b1, ExtREAL
for b5 being Function of RAT ,b2
for b6 being Real
for b7 being Element of b2 holds
( b3 is without-infty & b4 is without-infty & ( for b8 being Rational holds b5 . b8 = (b7 /\ (less_dom b3,(R_EAL b8))) /\ (b7 /\ (less_dom b4,(R_EAL (b6 - b8)))) ) implies b7 /\ (less_dom (b3 + b4),(R_EAL b6)) = union (rng b5) )
proof end;

definition
let c1 be non empty set ;
let c2 be PartFunc of c1, REAL ;
func R_EAL c2 -> PartFunc of a1, ExtREAL equals :: MESFUNC5:def 7
a2;
coherence
c2 is PartFunc of c1, ExtREAL
by PARTFUN1:31;
end;

:: deftheorem Def7 defines R_EAL MESFUNC5:def 7 :
for b1 being non empty set
for b2 being PartFunc of b1, REAL holds R_EAL b2 = b2;

theorem Th25: :: MESFUNC5:25
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( b4 is nonnegative & b5 is nonnegative implies b4 + b5 is nonnegative )
proof end;

theorem Th26: :: MESFUNC5:26
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being Real holds
( b2 is nonnegative implies ( ( 0 <= b3 implies b3 (#) b2 is nonnegative ) & ( b3 <= 0 implies b3 (#) b2 is nonpositive ) ) )
proof end;

theorem Th27: :: MESFUNC5:27
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL holds
( ( for b4 being set holds
( b4 in (dom b2) /\ (dom b3) implies ( b3 . b4 <= b2 . b4 & -infty < b3 . b4 & b2 . b4 < +infty ) ) ) implies b2 - b3 is nonnegative )
proof end;

Lemma31: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( max+ b4 is nonnegative & max- b4 is nonnegative & |.b4.| is nonnegative )
proof end;

theorem Th28: :: MESFUNC5:28
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL holds
( b2 is nonnegative & b3 is nonnegative implies ( dom (b2 + b3) = (dom b2) /\ (dom b3) & b2 + b3 is nonnegative ) )
proof end;

theorem Th29: :: MESFUNC5:29
for b1 being non empty set
for b2, b3, b4 being PartFunc of b1, ExtREAL holds
( b2 is nonnegative & b3 is nonnegative & b4 is nonnegative implies ( dom ((b2 + b3) + b4) = ((dom b2) /\ (dom b3)) /\ (dom b4) & (b2 + b3) + b4 is nonnegative & ( for b5 being set holds
( b5 in ((dom b2) /\ (dom b3)) /\ (dom b4) implies ((b2 + b3) + b4) . b5 = ((b2 . b5) + (b3 . b5)) + (b4 . b5) ) ) ) )
proof end;

theorem Th30: :: MESFUNC5:30
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL holds
( b2 is without-infty & b3 is without-infty implies ( 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 ) )
proof end;

theorem Th31: :: MESFUNC5:31
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL holds
( b2 is without-infty & b2 is without+infty & b3 is without-infty & b3 is without+infty implies ((max+ (b2 + b3)) + (max- b2)) + (max- b3) = ((max- (b2 + b3)) + (max+ b2)) + (max+ b3) )
proof end;

theorem Th32: :: MESFUNC5:32
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being Real holds
( 0 <= b3 implies ( max+ (b3 (#) b2) = b3 (#) (max+ b2) & max- (b3 (#) b2) = b3 (#) (max- b2) ) )
proof end;

theorem Th33: :: MESFUNC5:33
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being Real holds
( 0 <= b3 implies ( max+ ((- b3) (#) b2) = b3 (#) (max- b2) & max- ((- b3) (#) b2) = b3 (#) (max+ b2) ) )
proof end;

theorem Th34: :: MESFUNC5:34
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being set holds
( max+ (b2 | b3) = (max+ b2) | b3 & max- (b2 | b3) = (max- b2) | b3 )
proof end;

theorem Th35: :: MESFUNC5:35
for b1 being non empty set
for b2, b3 being PartFunc of b1, ExtREAL
for b4 being set holds
( b4 c= dom (b2 + b3) implies ( dom ((b2 + b3) | b4) = b4 & dom ((b2 | b4) + (b3 | b4)) = b4 & (b2 + b3) | b4 = (b2 | b4) + (b3 | b4) ) )
proof end;

theorem Th36: :: MESFUNC5:36
for b1 being non empty set
for b2 being PartFunc of b1, ExtREAL
for b3 being R_eal holds eq_dom b2,b3 = b2 " {b3}
proof end;

theorem Th37: :: MESFUNC5:37
for b1 being non empty set
for b2 being SigmaField of b1
for b3, b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 holds
( b3 is without-infty & b4 is without-infty & b3 is_measurable_on b5 & b4 is_measurable_on b5 implies b3 + b4 is_measurable_on b5 )
proof end;

Lemma42: for b1 being non empty set
for b2 being SigmaField of b1
for b3, b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 holds
( b3 is nonnegative & b4 is nonnegative & b3 is_measurable_on b5 & b4 is_measurable_on b5 implies ( dom (b3 + b4) = (dom b3) /\ (dom b4) & b3 + b4 is_measurable_on b5 ) )
proof end;

theorem Th38: :: MESFUNC5:38
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
not ( b4 is_simple_func_in b2 & dom b4 = {} & ( for b5 being Finite_Sep_Sequence of b2
for b6, b7 being FinSequence of ExtREAL holds
not ( b5,b6 are_Re-presentation_of b4 & b6 . 1 = 0 & ( for b8 being Nat holds
( 2 <= b8 & b8 in dom b6 implies ( 0 < b6 . b8 & b6 . b8 < +infty ) ) ) & dom b7 = dom b5 & ( for b8 being Nat holds
( b8 in dom b7 implies b7 . b8 = (b6 . b8) * ((b3 * b5) . b8) ) ) & Sum b7 = 0 ) ) )
proof end;

theorem Th39: :: MESFUNC5:39
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2
for b5, b6 being Real holds
( b3 is_measurable_on b4 & b4 c= dom b3 implies (b4 /\ (great_eq_dom b3,(R_EAL b5))) /\ (less_dom b3,(R_EAL b6)) is_measurable_on b2 )
proof end;

theorem Th40: :: MESFUNC5:40
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 holds
( b4 is_simple_func_in b2 implies b4 | b5 is_simple_func_in b2 )
proof end;

theorem Th41: :: MESFUNC5:41
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being Element of b2
for b4 being Finite_Sep_Sequence of b2
for b5 being FinSequence holds
( dom b4 = dom b5 & ( for b6 being Nat holds
( b6 in dom b4 implies b5 . b6 = (b4 . b6) /\ b3 ) ) implies b5 is Finite_Sep_Sequence of b2 )
proof end;

theorem Th42: :: MESFUNC5:42
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL
for b4 being Element of b2
for b5, b6 being Finite_Sep_Sequence of b2
for b7 being FinSequence of ExtREAL holds
( dom b5 = dom b6 & ( for b8 being Nat holds
( b8 in dom b5 implies b6 . b8 = (b5 . b8) /\ b4 ) ) & b5,b7 are_Re-presentation_of b3 implies b6,b7 are_Re-presentation_of b3 | b4 )
proof end;

theorem Th43: :: MESFUNC5:43
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( b4 is_simple_func_in b2 implies dom b4 is Element of b2 )
proof end;

Lemma48: for b1 being set
for b2 being FinSequence holds
( ( for b3 being Nat holds
( b3 in dom b2 implies b2 . b3 in b1 ) ) implies b2 is FinSequence of b1 )
proof end;

Lemma49: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( b4 is_simple_func_in b2 & dom b4 <> {} & b5 is_simple_func_in b2 & dom b5 = dom b4 implies ( b4 + b5 is_simple_func_in b2 & dom (b4 + b5) <> {} ) )
proof end;

theorem Th44: :: MESFUNC5:44
for b1 being non empty set
for b2 being SigmaField of b1
for b3, b4 being PartFunc of b1, ExtREAL holds
( b3 is_simple_func_in b2 & b4 is_simple_func_in b2 implies b3 + b4 is_simple_func_in b2 )
proof end;

theorem Th45: :: MESFUNC5:45
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real holds
( b4 is_simple_func_in b2 implies b5 (#) b4 is_simple_func_in b2 )
proof end;

theorem Th46: :: MESFUNC5:46
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( b4 is_simple_func_in b2 & b5 is_simple_func_in b2 & ( for b6 being set holds
( b6 in dom (b4 - b5) implies b5 . b6 <= b4 . b6 ) ) implies b4 - b5 is nonnegative )
proof end;

theorem Th47: :: MESFUNC5:47
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being Element of b2
for b5 being R_eal holds
not ( b5 <> +infty & b5 <> -infty & ( for b6 being PartFunc of b1, ExtREAL holds
not ( b6 is_simple_func_in b2 & dom b6 = b4 & ( for b7 being set holds
( b7 in b4 implies b6 . b7 = b5 ) ) ) ) )
proof end;

theorem Th48: :: MESFUNC5:48
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 holds
( b4 is_measurable_on b5 & b6 = (dom b4) /\ b5 implies b4 | b5 is_measurable_on b6 )
proof end;

theorem Th49: :: MESFUNC5:49
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being Element of b2
for b5, b6 being PartFunc of b1, ExtREAL holds
( b4 c= dom b5 & b5 is_measurable_on b4 & b6 is_measurable_on b4 & b5 is without-infty & b6 is without-infty implies (max+ (b5 + b6)) + (max- b5) is_measurable_on b4 )
proof end;

theorem Th50: :: MESFUNC5:50
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being Element of b2
for b5, b6 being PartFunc of b1, ExtREAL holds
( b4 c= (dom b5) /\ (dom b6) & b5 is_measurable_on b4 & b6 is_measurable_on b4 & b5 is without-infty & b6 is without-infty implies (max- (b5 + b6)) + (max+ b5) is_measurable_on b4 )
proof end;

theorem Th51: :: MESFUNC5:51
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being set holds
( b4 in b2 implies 0 <= b3 . b4 )
proof end;

Lemma58: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real holds
( dom b4 in b2 & ( for b6 being set holds
( b6 in dom b4 implies b4 . b6 = b5 ) ) implies b4 is_simple_func_in b2 )
proof end;

theorem Th52: :: MESFUNC5:52
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( 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 implies dom (b4 + b5) in b2 )
proof end;

Lemma60: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being Element of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being real number holds b3 /\ (less_dom b4,(R_EAL b5)) = less_dom (b4 | b3),(R_EAL b5)
proof end;

Lemma61: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being Element of b2
for b5 being PartFunc of b1, ExtREAL holds
( b5 | b4 is_measurable_on b4 iff b5 is_measurable_on b4 )
proof end;

Lemma62: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
not ( 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 ) & dom b4 = dom b5 & ( for b6 being Element of b2 holds
not ( b6 = dom (b4 + b5) & b4 + b5 is_measurable_on b6 ) ) )
proof end;

theorem Th53: :: MESFUNC5:53
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
not ( 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 ) & ( for b6 being Element of b2 holds
not ( b6 = dom (b4 + b5) & b4 + b5 is_measurable_on b6 ) ) )
proof end;

theorem Th54: :: MESFUNC5:54
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 holds
( dom b4 = b5 implies ( b4 is_measurable_on b6 iff b4 is_measurable_on b5 /\ b6 ) )
proof end;

theorem Th55: :: MESFUNC5:55
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( ex b5 being Element of b2 st dom b4 = b5 implies for b5 being Real
for b6 being Element of b2 holds
( b4 is_measurable_on b6 implies b5 (#) b4 is_measurable_on b6 ) )
proof end;

definition
mode ExtREAL_sequence is Function of NAT , ExtREAL ;
end;

definition
let c1 be ExtREAL_sequence;
attr a1 is convergent_to_finite_number means :Def8: :: MESFUNC5:def 8
ex b1 being real number st
for b2 being real number holds
not ( 0 < b2 & ( for b3 being Nat holds
ex b4 being Nat st
( b3 <= b4 & not |.((a1 . b4) - (R_EAL b1)).| < b2 ) ) );
end;

:: deftheorem Def8 defines convergent_to_finite_number MESFUNC5:def 8 :
for b1 being ExtREAL_sequence holds
( b1 is convergent_to_finite_number iff ex b2 being real number st
for b3 being real number holds
not ( 0 < b3 & ( for b4 being Nat holds
ex b5 being Nat st
( b4 <= b5 & not |.((b1 . b5) - (R_EAL b2)).| < b3 ) ) ) );

definition
let c1 be ExtREAL_sequence;
attr a1 is convergent_to_+infty means :Def9: :: MESFUNC5:def 9
for b1 being real number holds
not ( 0 < b1 & ( for b2 being Nat holds
ex b3 being Nat st
( b2 <= b3 & not b1 <= a1 . b3 ) ) );
end;

:: deftheorem Def9 defines convergent_to_+infty MESFUNC5:def 9 :
for b1 being ExtREAL_sequence holds
( b1 is convergent_to_+infty iff for b2 being real number holds
not ( 0 < b2 & ( for b3 being Nat holds
ex b4 being Nat st
( b3 <= b4 & not b2 <= b1 . b4 ) ) ) );

definition
let c1 be ExtREAL_sequence;
attr a1 is convergent_to_-infty means :Def10: :: MESFUNC5:def 10
for b1 being real number holds
not ( b1 < 0 & ( for b2 being Nat holds
ex b3 being Nat st
( b2 <= b3 & not a1 . b3 <= b1 ) ) );
end;

:: deftheorem Def10 defines convergent_to_-infty MESFUNC5:def 10 :
for b1 being ExtREAL_sequence holds
( b1 is convergent_to_-infty iff for b2 being real number holds
not ( b2 < 0 & ( for b3 being Nat holds
ex b4 being Nat st
( b3 <= b4 & not b1 . b4 <= b2 ) ) ) );

theorem Th56: :: MESFUNC5:56
for b1 being ExtREAL_sequence holds
( b1 is convergent_to_+infty implies ( not b1 is convergent_to_-infty & not b1 is convergent_to_finite_number ) )
proof end;

theorem Th57: :: MESFUNC5:57
for b1 being ExtREAL_sequence holds
( b1 is convergent_to_-infty implies ( not b1 is convergent_to_+infty & not b1 is convergent_to_finite_number ) )
proof end;

definition
let c1 be ExtREAL_sequence;
attr a1 is convergent means :Def11: :: MESFUNC5:def 11
not ( not a1 is convergent_to_finite_number & not a1 is convergent_to_+infty & not a1 is convergent_to_-infty );
end;

:: deftheorem Def11 defines convergent MESFUNC5:def 11 :
for b1 being ExtREAL_sequence holds
( b1 is convergent iff not ( not b1 is convergent_to_finite_number & not b1 is convergent_to_+infty & not b1 is convergent_to_-infty ) );

definition
let c1 be ExtREAL_sequence;
assume E72: c1 is convergent ;
func lim c1 -> R_eal means :Def12: :: MESFUNC5:def 12
not ( ( for b1 being real number holds
not ( a2 = b1 & ( for b2 being real number holds
not ( 0 < b2 & ( for b3 being Nat holds
ex b4 being Nat st
( b3 <= b4 & not |.((a1 . b4) - a2).| < b2 ) ) ) ) & a1 is convergent_to_finite_number ) ) & not ( a2 = +infty & a1 is convergent_to_+infty ) & not ( a2 = -infty & a1 is convergent_to_-infty ) );
existence
not for b1 being R_eal holds
( ( for b2 being real number holds
not ( b1 = b2 & ( for b3 being real number holds
not ( 0 < b3 & ( for b4 being Nat holds
ex b5 being Nat st
( b4 <= b5 & not |.((c1 . b5) - b1).| < b3 ) ) ) ) & c1 is convergent_to_finite_number ) ) & not ( b1 = +infty & c1 is convergent_to_+infty ) & not ( b1 = -infty & c1 is convergent_to_-infty ) )
proof end;
uniqueness
for b1, b2 being R_eal holds
( not ( ( for b3 being real number holds
not ( b1 = b3 & ( for b4 being real number holds
not ( 0 < b4 & ( for b5 being Nat holds
ex b6 being Nat st
( b5 <= b6 & not |.((c1 . b6) - b1).| < b4 ) ) ) ) & c1 is convergent_to_finite_number ) ) & not ( b1 = +infty & c1 is convergent_to_+infty ) & not ( b1 = -infty & c1 is convergent_to_-infty ) ) & not ( ( for b3 being real number holds
not ( b2 = b3 & ( for b4 being real number holds
not ( 0 < b4 & ( for b5 being Nat holds
ex b6 being Nat st
( b5 <= b6 & not |.((c1 . b6) - b2).| < b4 ) ) ) ) & c1 is convergent_to_finite_number ) ) & not ( b2 = +infty & c1 is convergent_to_+infty ) & not ( b2 = -infty & c1 is convergent_to_-infty ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def12 defines lim MESFUNC5:def 12 :
for b1 being ExtREAL_sequence holds
( b1 is convergent implies for b2 being R_eal holds
( b2 = lim b1 iff not ( ( for b3 being real number holds
not ( b2 = b3 & ( for b4 being real number holds
not ( 0 < b4 & ( for b5 being Nat holds
ex b6 being Nat st
( b5 <= b6 & not |.((b1 . b6) - b2).| < b4 ) ) ) ) & b1 is convergent_to_finite_number ) ) & not ( b2 = +infty & b1 is convergent_to_+infty ) & not ( b2 = -infty & b1 is convergent_to_-infty ) ) ) );

theorem Th58: :: MESFUNC5:58
for b1 being ExtREAL_sequence
for b2 being real number holds
( ( for b3 being Nat holds b1 . b3 = b2 ) implies ( b1 is convergent_to_finite_number & lim b1 = b2 ) )
proof end;

theorem Th59: :: MESFUNC5:59
for b1 being FinSequence of ExtREAL holds
( ( for b2 being Nat holds
( b2 in dom b1 implies 0 <= b1 . b2 ) ) implies 0 <= Sum b1 )
proof end;

theorem Th60: :: MESFUNC5:60
for b1 being ExtREAL_sequence holds
( ( for b2, b3 being Nat holds
( b2 <= b3 implies b1 . b2 <= b1 . b3 ) ) implies ( b1 is convergent & lim b1 = sup (rng b1) ) )
proof end;

theorem Th61: :: MESFUNC5:61
for b1, b2 being ExtREAL_sequence holds
( ( for b3 being Nat holds b1 . b3 <= b2 . b3 ) implies sup (rng b1) <= sup (rng b2) )
proof end;

theorem Th62: :: MESFUNC5:62
for b1 being ExtREAL_sequence
for b2 being Nat holds b1 . b2 <= sup (rng b1)
proof end;

theorem Th63: :: MESFUNC5:63
for b1 being ExtREAL_sequence
for b2 being R_eal holds
( ( for b3 being Nat holds b1 . b3 <= b2 ) implies sup (rng b1) <= b2 )
proof end;

theorem Th64: :: MESFUNC5:64
for b1 being ExtREAL_sequence
for b2 being R_eal holds
not ( b2 <> +infty & ( for b3 being Nat holds b1 . b3 <= b2 ) & not sup (rng b1) < +infty )
proof end;

theorem Th65: :: MESFUNC5:65
for b1 being ExtREAL_sequence holds
( b1 is without-infty implies ( not ( sup (rng b1) <> +infty & ( for b2 being real number holds
not ( 0 < b2 & ( for b3 being Nat holds b1 . b3 <= b2 ) ) ) ) & not ( ex b2 being real number st
( 0 < b2 & ( for b3 being Nat holds b1 . b3 <= b2 ) ) & not sup (rng b1) <> +infty ) ) )
proof end;

theorem Th66: :: MESFUNC5:66
for b1 being ExtREAL_sequence
for b2 being R_eal holds
( ( for b3 being Nat holds b1 . b3 = b2 ) implies ( b1 is convergent & lim b1 = b2 & lim b1 = sup (rng b1) ) )
proof end;

Lemma81: for b1 being ExtREAL_sequence holds
not ( b1 is without-infty & not sup (rng b1) in REAL & not sup (rng b1) = +infty )
proof end;

theorem Th67: :: MESFUNC5:67
for b1, b2, b3 being ExtREAL_sequence holds
( ( for b4, b5 being Nat holds
( b4 <= b5 implies b1 . b4 <= b1 . b5 ) ) & ( for b4, b5 being Nat holds
( b4 <= b5 implies b2 . b4 <= b2 . b5 ) ) & b1 is without-infty & b2 is without-infty & ( for b4 being Nat holds (b1 . b4) + (b2 . b4) = b3 . b4 ) implies ( b3 is convergent & lim b3 = sup (rng b3) & lim b3 = (lim b1) + (lim b2) & sup (rng b3) = (sup (rng b2)) + (sup (rng b1)) ) )
proof end;

theorem Th68: :: MESFUNC5:68
for b1, b2 being ExtREAL_sequence
for b3 being Real holds
( 0 <= b3 & b1 is without-infty & ( for b4 being Nat holds b2 . b4 = (R_EAL b3) * (b1 . b4) ) implies ( sup (rng b2) = (R_EAL b3) * (sup (rng b1)) & b2 is without-infty ) )
proof end;

theorem Th69: :: MESFUNC5:69
for b1, b2 being ExtREAL_sequence
for b3 being Real holds
( 0 <= b3 & ( for b4, b5 being Nat holds
( b4 <= b5 implies b1 . b4 <= b1 . b5 ) ) & ( for b4 being Nat holds b2 . b4 = (R_EAL b3) * (b1 . b4) ) & b1 is without-infty implies ( ( for b4, b5 being Nat holds
( b4 <= b5 implies b2 . b4 <= b2 . b5 ) ) & b2 is without-infty & b2 is convergent & lim b2 = sup (rng b2) & lim b2 = (R_EAL b3) * (lim b1) ) )
proof end;

definition
let c1 be non empty set ;
let c2 be Functional_Sequence of c1, ExtREAL ;
let c3 be Element of c1;
func c2 # c3 -> ExtREAL_sequence means :Def13: :: MESFUNC5:def 13
for b1 being Nat holds a4 . b1 = (a2 . b1) . a3;
existence
ex b1 being ExtREAL_sequence st
for b2 being Nat holds b1 . b2 = (c2 . b2) . c3
proof end;
uniqueness
for b1, b2 being ExtREAL_sequence holds
( ( for b3 being Nat holds b1 . b3 = (c2 . b3) . c3 ) & ( for b3 being Nat holds b2 . b3 = (c2 . b3) . c3 ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def13 defines # MESFUNC5:def 13 :
for b1 being non empty set
for b2 being Functional_Sequence of b1, ExtREAL
for b3 being Element of b1
for b4 being ExtREAL_sequence holds
( b4 = b2 # b3 iff for b5 being Nat holds b4 . b5 = (b2 . b5) . b3 );

definition
let c1, c2 be set ;
let c3 be Function of NAT , PFuncs c1,c2;
let c4 be Nat;
redefine func . as c3 . c4 -> PartFunc of a1,a2;
coherence
c3 . c4 is PartFunc of c1,c2
proof end;
end;

theorem Th70: :: MESFUNC5:70
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being PartFunc of b1, ExtREAL holds
not ( ex b4 being Element of b2 st
( b4 = dom b3 & b3 is_measurable_on b4 ) & b3 is nonnegative & ( for b4 being Functional_Sequence of b1, ExtREAL holds
not ( ( for b5 being Nat holds
( b4 . b5 is_simple_func_in b2 & dom (b4 . b5) = dom b3 ) ) & ( for b5 being Nat holds b4 . b5 is nonnegative ) & ( for b5, b6 being Nat holds
( b5 <= b6 implies for b7 being Element of b1 holds
( b7 in dom b3 implies (b4 . b5) . b7 <= (b4 . b6) . b7 ) ) ) & ( for b5 being Element of b1 holds
( b5 in dom b3 implies ( b4 # b5 is convergent & lim (b4 # b5) = b3 . b5 ) ) ) ) ) )
proof end;

definition
let c1 be non empty set ;
let c2 be SigmaField of c1;
let c3 be sigma_Measure of c2;
let c4 be PartFunc of c1, ExtREAL ;
func integral' c3,c4 -> Element of ExtREAL equals :Def14: :: MESFUNC5:def 14
integral a1,a2,a3,a4 if dom a4 <> {}
otherwise 0. ;
correctness
coherence
( ( dom c4 <> {} implies integral c1,c2,c3,c4 is Element of ExtREAL ) & ( not dom c4 <> {} implies 0. is Element of ExtREAL ) )
;
consistency
for b1 being Element of ExtREAL holds
verum
;
;
end;

:: deftheorem Def14 defines integral' MESFUNC5:def 14 :
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( ( dom b4 <> {} implies integral' b3,b4 = integral b1,b2,b3,b4 ) & ( not dom b4 <> {} implies integral' b3,b4 = 0. ) );

theorem Th71: :: MESFUNC5:71
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( b4 is_simple_func_in b2 & b5 is_simple_func_in b2 & b4 is nonnegative & b5 is nonnegative implies ( dom (b4 + b5) = (dom b4) /\ (dom b5) & integral' b3,(b4 + b5) = (integral' b3,(b4 | (dom (b4 + b5)))) + (integral' b3,(b5 | (dom (b4 + b5)))) ) )
proof end;

theorem Th72: :: MESFUNC5:72
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real holds
( b4 is_simple_func_in b2 & b4 is nonnegative & 0 <= b5 implies integral' b3,(b5 (#) b4) = (R_EAL b5) * (integral' b3,b4) )
proof end;

theorem Th73: :: MESFUNC5:73
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 holds
( b4 is_simple_func_in b2 & b4 is nonnegative & b5 misses b6 implies integral' b3,(b4 | (b5 \/ b6)) = (integral' b3,(b4 | b5)) + (integral' b3,(b4 | b6)) )
proof end;

theorem Th74: :: MESFUNC5:74
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( b4 is_simple_func_in b2 & b4 is nonnegative implies 0 <= integral' b3,b4 )
proof end;

Lemma92: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( b4 is_simple_func_in b2 & dom b4 <> {} & b4 is nonnegative & b5 is_simple_func_in b2 & dom b5 = dom b4 & b5 is nonnegative & ( for b6 being set holds
( b6 in dom b4 implies b5 . b6 <= b4 . b6 ) ) implies ( b4 - b5 is_simple_func_in b2 & dom (b4 - b5) <> {} & b4 - b5 is nonnegative & integral b1,b2,b3,b4 = (integral b1,b2,b3,(b4 - b5)) + (integral b1,b2,b3,b5) ) )
proof end;

theorem Th75: :: MESFUNC5:75
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( b4 is_simple_func_in b2 & b4 is nonnegative & b5 is_simple_func_in b2 & b5 is nonnegative & ( for b6 being set holds
( b6 in dom (b4 - b5) implies b5 . b6 <= b4 . b6 ) ) implies ( dom (b4 - b5) = (dom b4) /\ (dom b5) & integral' b3,(b4 | (dom (b4 - b5))) = (integral' b3,(b4 - b5)) + (integral' b3,(b5 | (dom (b4 - b5)))) ) )
proof end;

theorem Th76: :: MESFUNC5:76
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( b4 is_simple_func_in b2 & b5 is_simple_func_in b2 & b4 is nonnegative & b5 is nonnegative & ( for b6 being set holds
( b6 in dom (b4 - b5) implies b5 . b6 <= b4 . b6 ) ) implies integral' b3,(b5 | (dom (b4 - b5))) <= integral' b3,(b4 | (dom (b4 - b5))) )
proof end;

theorem Th77: :: MESFUNC5:77
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being R_eal holds
( 0 <= b5 & b4 is_simple_func_in b2 & ( for b6 being set holds
( b6 in dom b4 implies b4 . b6 = b5 ) ) implies integral' b3,b4 = b5 * (b3 . (dom b4)) )
proof end;

theorem Th78: :: MESFUNC5:78
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( b4 is_simple_func_in b2 & b4 is nonnegative implies integral' b3,(b4 | (eq_dom b4,(R_EAL 0))) = 0 )
proof end;

theorem Th79: :: MESFUNC5:79
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being Element of b2
for b5 being PartFunc of b1, ExtREAL holds
( b5 is_simple_func_in b2 & b3 . b4 = 0 & b5 is nonnegative implies integral' b3,(b5 | b4) = 0 )
proof end;

theorem Th80: :: MESFUNC5:80
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Functional_Sequence of b1, ExtREAL
for b6 being ExtREAL_sequence holds
( b4 is_simple_func_in b2 & ( for b7 being set holds
not ( b7 in dom b4 & not 0 < b4 . b7 ) ) & ( for b7 being Nat holds b5 . b7 is_simple_func_in b2 ) & ( for b7 being Nat holds dom (b5 . b7) = dom b4 ) & ( for b7 being Nat holds b5 . b7 is nonnegative ) & ( for b7, b8 being Nat holds
( b7 <= b8 implies for b9 being Element of b1 holds
( b9 in dom b4 implies (b5 . b7) . b9 <= (b5 . b8) . b9 ) ) ) & ( for b7 being Element of b1 holds
( b7 in dom b4 implies ( b5 # b7 is convergent & b4 . b7 <= lim (b5 # b7) ) ) ) & ( for b7 being Nat holds b6 . b7 = integral' b3,(b5 . b7) ) implies ( b6 is convergent & integral' b3,b4 <= lim b6 ) )
proof end;

theorem Th81: :: MESFUNC5:81
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Functional_Sequence of b1, ExtREAL holds
not ( b4 is_simple_func_in b2 & b4 is nonnegative & ( for b6 being Nat holds b5 . b6 is_simple_func_in b2 ) & ( for b6 being Nat holds dom (b5 . b6) = dom b4 ) & ( for b6 being Nat holds b5 . b6 is nonnegative ) & ( for b6, b7 being Nat holds
( b6 <= b7 implies for b8 being Element of b1 holds
( b8 in dom b4 implies (b5 . b6) . b8 <= (b5 . b7) . b8 ) ) ) & ( for b6 being Element of b1 holds
( b6 in dom b4 implies ( b5 # b6 is convergent & b4 . b6 <= lim (b5 # b6) ) ) ) & ( for b6 being ExtREAL_sequence holds
not ( ( for b7 being Nat holds b6 . b7 = integral' b3,(b5 . b7) ) & b6 is convergent & sup (rng b6) = lim b6 & integral' b3,b4 <= lim b6 ) ) )
proof end;

theorem Th82: :: MESFUNC5:82
for b1 being non empty set
for b2 being SigmaField of 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 ExtREAL_sequence holds
( ( for b9 being Nat holds
( b5 . b9 is_simple_func_in b2 & dom (b5 . b9) = b4 ) ) & ( for b9 being Nat holds b5 . b9 is nonnegative ) & ( for b9, b10 being Nat holds
( b9 <= b10 implies for b11 being Element of b1 holds
( b11 in b4 implies (b5 . b9) . b11 <= (b5 . b10) . b11 ) ) ) & ( for b9 being Nat holds
( b6 . b9 is_simple_func_in b2 & dom (b6 . b9) = b4 ) ) & ( for b9 being Nat holds b6 . b9 is nonnegative ) & ( for b9, b10 being Nat holds
( b9 <= b10 implies for b11 being Element of b1 holds
( b11 in b4 implies (b6 . b9) . b11 <= (b6 . b10) . b11 ) ) ) & ( for b9 being Element of b1 holds
( b9 in b4 implies ( b5 # b9 is convergent & b6 # b9 is convergent & lim (b5 # b9) = lim (b6 # b9) ) ) ) & ( for b9 being Nat holds
( b7 . b9 = integral' b3,(b5 . b9) & b8 . b9 = integral' b3,(b6 . b9) ) ) implies ( b7 is convergent & b8 is convergent & lim b7 = lim b8 ) )
proof end;

definition
let c1 be non empty set ;
let c2 be SigmaField of c1;
let c3 be sigma_Measure of c2;
let c4 be PartFunc of c1, ExtREAL ;
assume that
E101: ex b1 being Element of c2 st
( b1 = dom c4 & c4 is_measurable_on b1 ) and
E102: c4 is nonnegative ;
func integral+ c3,c4 -> Element of ExtREAL means :Def15: :: MESFUNC5:def 15
ex b1 being Functional_Sequence of a1, ExtREAL ex b2 being ExtREAL_sequence st
( ( for b3 being Nat holds
( b1 . b3 is_simple_func_in a2 & dom (b1 . b3) = dom a4 ) ) & ( for b3 being Nat holds b1 . b3 is nonnegative ) & ( for b3, b4 being Nat holds
( b3 <= b4 implies for b5 being Element of a1 holds
( b5 in dom a4 implies (b1 . b3) . b5 <= (b1 . b4) . b5 ) ) ) & ( for b3 being Element of a1 holds
( b3 in dom a4 implies ( b1 # b3 is convergent & lim (b1 # b3) = a4 . b3 ) ) ) & ( for b3 being Nat holds b2 . b3 = integral' a3,(b1 . b3) ) & b2 is convergent & a5 = lim b2 );
existence
ex b1 being Element of ExtREAL ex b2 being Functional_Sequence of c1, ExtREAL ex b3 being ExtREAL_sequence st
( ( for b4 being Nat holds
( b2 . b4 is_simple_func_in c2 & dom (b2 . b4) = dom c4 ) ) & ( for b4 being Nat holds b2 . b4 is nonnegative ) & ( for b4, b5 being Nat holds
( b4 <= b5 implies for b6 being Element of c1 holds
( b6 in dom c4 implies (b2 . b4) . b6 <= (b2 . b5) . b6 ) ) ) & ( for b4 being Element of c1 holds
( b4 in dom c4 implies ( b2 # b4 is convergent & lim (b2 # b4) = c4 . b4 ) ) ) & ( for b4 being Nat holds b3 . b4 = integral' c3,(b2 . b4) ) & b3 is convergent & b1 = lim b3 )
proof end;
uniqueness
for b1, b2 being Element of ExtREAL holds
( ex b3 being Functional_Sequence of c1, ExtREAL ex b4 being ExtREAL_sequence st
( ( for b5 being Nat holds
( b3 . b5 is_simple_func_in c2 & dom (b3 . b5) = dom c4 ) ) & ( for b5 being Nat holds b3 . b5 is nonnegative ) & ( for b5, b6 being Nat holds
( b5 <= b6 implies for b7 being Element of c1 holds
( b7 in dom c4 implies (b3 . b5) . b7 <= (b3 . b6) . b7 ) ) ) & ( for b5 being Element of c1 holds
( b5 in dom c4 implies ( b3 # b5 is convergent & lim (b3 # b5) = c4 . b5 ) ) ) & ( for b5 being Nat holds b4 . b5 = integral' c3,(b3 . b5) ) & b4 is convergent & b1 = lim b4 ) & ex b3 being Functional_Sequence of c1, ExtREAL ex b4 being ExtREAL_sequence st
( ( for b5 being Nat holds
( b3 . b5 is_simple_func_in c2 & dom (b3 . b5) = dom c4 ) ) & ( for b5 being Nat holds b3 . b5 is nonnegative ) & ( for b5, b6 being Nat holds
( b5 <= b6 implies for b7 being Element of c1 holds
( b7 in dom c4 implies (b3 . b5) . b7 <= (b3 . b6) . b7 ) ) ) & ( for b5 being Element of c1 holds
( b5 in dom c4 implies ( b3 # b5 is convergent & lim (b3 # b5) = c4 . b5 ) ) ) & ( for b5 being Nat holds b4 . b5 = integral' c3,(b3 . b5) ) & b4 is convergent & b2 = lim b4 ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def15 defines integral+ MESFUNC5:def 15 :
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) & b4 is nonnegative implies for b5 being Element of ExtREAL holds
( b5 = integral+ b3,b4 iff ex b6 being Functional_Sequence of b1, ExtREAL ex b7 being ExtREAL_sequence st
( ( for b8 being Nat holds
( b6 . b8 is_simple_func_in b2 & dom (b6 . b8) = dom b4 ) ) & ( for b8 being Nat holds b6 . b8 is nonnegative ) & ( for b8, b9 being Nat holds
( b8 <= b9 implies for b10 being Element of b1 holds
( b10 in dom b4 implies (b6 . b8) . b10 <= (b6 . b9) . b10 ) ) ) & ( for b8 being Element of b1 holds
( b8 in dom b4 implies ( b6 # b8 is convergent & lim (b6 # b8) = b4 . b8 ) ) ) & ( for b8 being Nat holds b7 . b8 = integral' b3,(b6 . b8) ) & b7 is convergent & b5 = lim b7 ) ) );

theorem Th83: :: MESFUNC5:83
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( b4 is_simple_func_in b2 & b4 is nonnegative implies integral+ b3,b4 = integral' b3,b4 )
proof end;

Lemma103: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( 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 implies integral+ b3,(b4 + b5) = (integral+ b3,b4) + (integral+ b3,b5) )
proof end;

theorem Th84: :: MESFUNC5:84
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
not ( 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 & ( for b6 being Element of b2 holds
not ( b6 = dom (b4 + b5) & integral+ b3,(b4 + b5) = (integral+ b3,(b4 | b6)) + (integral+ b3,(b5 | b6)) ) ) )
proof end;

theorem Th85: :: MESFUNC5:85
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) & b4 is nonnegative implies 0 <= integral+ b3,b4 )
proof end;

theorem Th86: :: MESFUNC5:86
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 holds
( ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & b4 is nonnegative implies 0 <= integral+ b3,(b4 | b5) )
proof end;

theorem Th87: :: MESFUNC5:87
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 holds
( ex b7 being Element of b2 st
( b7 = dom b4 & b4 is_measurable_on b7 ) & b4 is nonnegative & b5 misses b6 implies integral+ b3,(b4 | (b5 \/ b6)) = (integral+ b3,(b4 | b5)) + (integral+ b3,(b4 | b6)) )
proof end;

theorem Th88: :: MESFUNC5:88
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 holds
( ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & b4 is nonnegative & b3 . b5 = 0 implies integral+ b3,(b4 | b5) = 0 )
proof end;

theorem Th89: :: MESFUNC5:89
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 holds
( ex b7 being Element of b2 st
( b7 = dom b4 & b4 is_measurable_on b7 ) & b4 is nonnegative & b5 c= b6 implies integral+ b3,(b4 | b5) <= integral+ b3,(b4 | b6) )
proof end;

theorem Th90: :: MESFUNC5:90
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 holds
( b4 is nonnegative & b5 = dom b4 & b4 is_measurable_on b5 & b3 . b6 = 0 implies integral+ b3,(b4 | (b5 \ b6)) = integral+ b3,b4 )
proof end;

theorem Th91: :: MESFUNC5:91
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( 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 holds
( b6 in dom b5 implies b5 . b6 <= b4 . b6 ) ) implies integral+ b3,b5 <= integral+ b3,b4 )
proof end;

theorem Th92: :: MESFUNC5:92
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real holds
( 0 <= b5 & ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & b4 is nonnegative implies integral+ b3,(b5 (#) b4) = (R_EAL b5) * (integral+ b3,b4) )
proof end;

theorem Th93: :: MESFUNC5:93
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) & ( for b5 being Element of b1 holds
( b5 in dom b4 implies 0 = b4 . b5 ) ) implies integral+ b3,b4 = 0 )
proof end;

definition
let c1 be non empty set ;
let c2 be SigmaField of c1;
let c3 be sigma_Measure of c2;
let c4 be PartFunc of c1, ExtREAL ;
func Integral c3,c4 -> Element of ExtREAL equals :: MESFUNC5:def 16
(integral+ a3,(max+ a4)) - (integral+ a3,(max- a4));
coherence
(integral+ c3,(max+ c4)) - (integral+ c3,(max- c4)) is Element of ExtREAL
;
end;

:: deftheorem Def16 defines Integral MESFUNC5:def 16 :
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds Integral b3,b4 = (integral+ b3,(max+ b4)) - (integral+ b3,(max- b4));

theorem Th94: :: MESFUNC5:94
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) & b4 is nonnegative implies Integral b3,b4 = integral+ b3,b4 )
proof end;

theorem Th95: :: MESFUNC5:95
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( b4 is_simple_func_in b2 & b4 is nonnegative implies ( Integral b3,b4 = integral+ b3,b4 & Integral b3,b4 = integral' b3,b4 ) )
proof end;

theorem Th96: :: MESFUNC5:96
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) & b4 is nonnegative implies 0 <= Integral b3,b4 )
proof end;

theorem Th97: :: MESFUNC5:97
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 holds
( ex b7 being Element of b2 st
( b7 = dom b4 & b4 is_measurable_on b7 ) & b4 is nonnegative & b5 misses b6 implies Integral b3,(b4 | (b5 \/ b6)) = (Integral b3,(b4 | b5)) + (Integral b3,(b4 | b6)) )
proof end;

theorem Th98: :: MESFUNC5:98
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 holds
( ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & b4 is nonnegative implies 0 <= Integral b3,(b4 | b5) )
proof end;

theorem Th99: :: MESFUNC5:99
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 holds
( ex b7 being Element of b2 st
( b7 = dom b4 & b4 is_measurable_on b7 ) & b4 is nonnegative & b5 c= b6 implies Integral b3,(b4 | b5) <= Integral b3,(b4 | b6) )
proof end;

theorem Th100: :: MESFUNC5:100
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 holds
( ex b6 being Element of b2 st
( b6 = dom b4 & b4 is_measurable_on b6 ) & b3 . b5 = 0 implies Integral b3,(b4 | b5) = 0 )
proof end;

theorem Th101: :: MESFUNC5:101
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 holds
( b5 = dom b4 & b4 is_measurable_on b5 & b3 . b6 = 0 implies Integral b3,(b4 | (b5 \ b6)) = Integral b3,b4 )
proof end;

definition
let c1 be non empty set ;
let c2 be SigmaField of c1;
let c3 be sigma_Measure of c2;
let c4 be PartFunc of c1, ExtREAL ;
pred c4 is_integrable_on c3 means :Def17: :: MESFUNC5:def 17
( 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;

:: deftheorem Def17 defines is_integrable_on MESFUNC5:def 17 :
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc 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 ) );

theorem Th102: :: MESFUNC5:102
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( b4 is_integrable_on b3 implies ( 0 <= integral+ b3,(max+ b4) & 0 <= integral+ b3,(max- b4) & -infty < Integral b3,b4 & Integral b3,b4 < +infty ) )
proof end;

theorem Th103: :: MESFUNC5:103
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 holds
( b4 is_integrable_on b3 implies ( integral+ b3,(max+ (b4 | b5)) <= integral+ b3,(max+ b4) & integral+ b3,(max- (b4 | b5)) <= integral+ b3,(max- b4) & b4 | b5 is_integrable_on b3 ) )
proof end;

theorem Th104: :: MESFUNC5:104
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 holds
( b4 is_integrable_on b3 & b5 misses b6 implies Integral b3,(b4 | (b5 \/ b6)) = (Integral b3,(b4 | b5)) + (Integral b3,(b4 | b6)) )
proof end;

theorem Th105: :: MESFUNC5:105
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5, b6 being Element of b2 holds
( b4 is_integrable_on b3 & b6 = (dom b4) \ b5 implies ( b4 | b5 is_integrable_on b3 & Integral b3,b4 = (Integral b3,(b4 | b5)) + (Integral b3,(b4 | b6)) ) )
proof end;

theorem Th106: :: MESFUNC5:106
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( ex b5 being Element of b2 st
( b5 = dom b4 & b4 is_measurable_on b5 ) implies ( b4 is_integrable_on b3 iff |.b4.| is_integrable_on b3 ) )
proof end;

theorem Th107: :: MESFUNC5:107
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( b4 is_integrable_on b3 implies |.(Integral b3,b4).| <= Integral b3,|.b4.| )
proof end;

theorem Th108: :: MESFUNC5:108
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( 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 holds
( b6 in dom b4 implies |.(b4 . b6).| <= b5 . b6 ) ) implies ( b4 is_integrable_on b3 & Integral b3,|.b4.| <= Integral b3,b5 ) )
proof end;

theorem Th109: :: MESFUNC5:109
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real holds
( dom b4 in b2 & 0 <= b5 & dom b4 <> {} & ( for b6 being set holds
( b6 in dom b4 implies b4 . b6 = b5 ) ) implies integral b1,b2,b3,b4 = (R_EAL b5) * (b3 . (dom b4)) )
proof end;

theorem Th110: :: MESFUNC5:110
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real holds
( dom b4 in b2 & 0 <= b5 & ( for b6 being set holds
( b6 in dom b4 implies b4 . b6 = b5 ) ) implies integral' b3,b4 = (R_EAL b5) * (b3 . (dom b4)) )
proof end;

theorem Th111: :: MESFUNC5:111
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL holds
( b4 is_integrable_on b3 implies ( 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 ) )
proof end;

theorem Th112: :: MESFUNC5:112
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( b4 is_integrable_on b3 & b5 is_integrable_on b3 & b4 is nonnegative & b5 is nonnegative implies b4 + b5 is_integrable_on b3 )
proof end;

theorem Th113: :: MESFUNC5:113
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( b4 is_integrable_on b3 & b5 is_integrable_on b3 implies dom (b4 + b5) in b2 )
proof end;

Lemma127: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( ex b6 being Element of b2 st
( dom (b4 + b5) = b6 & b4 + b5 is_measurable_on b6 ) & b4 is_integrable_on b3 & b5 is_integrable_on b3 implies b4 + b5 is_integrable_on b3 )
proof end;

theorem Th114: :: MESFUNC5:114
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( b4 is_integrable_on b3 & b5 is_integrable_on b3 implies b4 + b5 is_integrable_on b3 )
proof end;

Lemma129: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
not ( b4 is_integrable_on b3 & b5 is_integrable_on b3 & dom b4 = dom b5 & ( for b6, b7, b8 being Element of b2 holds
not ( b6 c= dom b4 & b7 c= dom b4 & b6 = (dom b4) \ b7 & b4 | b6 is_finite & b6 = dom (b4 | b6) & b4 | b6 is_measurable_on b6 & b4 | b6 is_integrable_on b3 & Integral b3,b4 = Integral b3,(b4 | b6) & b6 c= dom b5 & b7 c= dom b5 & b6 = (dom b5) \ b7 & b5 | b6 is_finite & b6 = dom (b5 | b6) & b5 | b6 is_measurable_on b6 & b5 | b6 is_integrable_on b3 & Integral b3,b5 = Integral b3,(b5 | b6) & b6 c= dom (b4 + b5) & b8 c= dom (b4 + b5) & b6 = (dom (b4 + b5)) \ b8 & b3 . b7 = 0 & b3 . b8 = 0 & b6 = dom ((b4 + b5) | b6) & (b4 + b5) | b6 is_measurable_on b6 & (b4 + b5) | b6 is_integrable_on b3 & (b4 + b5) | b6 = (b4 | b6) + (b5 | b6) & Integral b3,((b4 + b5) | b6) = (Integral b3,(b4 | b6)) + (Integral b3,(b5 | b6)) ) ) )
proof end;

Lemma130: for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
( b4 is_integrable_on b3 & b5 is_integrable_on b3 & dom b4 = dom b5 implies ( b4 + b5 is_integrable_on b3 & Integral b3,(b4 + b5) = (Integral b3,b4) + (Integral b3,b5) ) )
proof end;

theorem Th115: :: MESFUNC5:115
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL holds
not ( b4 is_integrable_on b3 & b5 is_integrable_on b3 & ( for b6 being Element of b2 holds
not ( b6 = (dom b4) /\ (dom b5) & Integral b3,(b4 + b5) = (Integral b3,(b4 | b6)) + (Integral b3,(b5 | b6)) ) ) )
proof end;

theorem Th116: :: MESFUNC5:116
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real holds
( b4 is_integrable_on b3 implies ( b5 (#) b4 is_integrable_on b3 & Integral b3,(b5 (#) b4) = (R_EAL b5) * (Integral b3,b4) ) )
proof end;

definition
let c1 be non empty set ;
let c2 be SigmaField of c1;
let c3 be sigma_Measure of c2;
let c4 be PartFunc of c1, ExtREAL ;
let c5 be Element of c2;
func Integral_on c3,c5,c4 -> Element of ExtREAL equals :: MESFUNC5:def 18
Integral a3,(a4 | a5);
coherence
Integral c3,(c4 | c5) is Element of ExtREAL
;
end;

:: deftheorem Def18 defines Integral_on MESFUNC5:def 18 :
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Element of b2 holds Integral_on b3,b5,b4 = Integral b3,(b4 | b5);

theorem Th117: :: MESFUNC5:117
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4, b5 being PartFunc of b1, ExtREAL
for b6 being Element of b2 holds
( b4 is_integrable_on b3 & b5 is_integrable_on b3 & b6 c= dom (b4 + b5) implies ( b4 + b5 is_integrable_on b3 & Integral_on b3,b6,(b4 + b5) = (Integral_on b3,b6,b4) + (Integral_on b3,b6,b5) ) )
proof end;

theorem Th118: :: MESFUNC5:118
for b1 being non empty set
for b2 being SigmaField of b1
for b3 being sigma_Measure of b2
for b4 being PartFunc of b1, ExtREAL
for b5 being Real
for b6 being Element of b2 holds
( b4 is_integrable_on b3 & b4 is_measurable_on b6 implies ( b4 | b6 is_integrable_on b3 & Integral_on b3,b6,(b5 (#) b4) = (R_EAL b5) * (Integral_on b3,b6,b4) ) )
proof end;