%   ORIGINAL: h4/lebesgue/pos__fn__integral__sum
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/lebesgue/pos__simple__fn__integral__sum: !x s m f a P. h4/measure/measure__space m /\ (!i. h4/bool/IN i P ==> h4/measure/pos__simple__fn m (f i) s a (x i)) /\ h4/pred__set/FINITE P /\ ~(P = h4/pred__set/EMPTY) ==> h4/measure/pos__simple__fn m (\t. h4/extreal/EXTREAL__SUM__IMAGE (\i. f i t) P) s a (\i. h4/real__sigma/REAL__SUM__IMAGE (\j. x j i) P) /\ h4/lebesgue/pos__simple__fn__integral m s a (\j. h4/real__sigma/REAL__SUM__IMAGE (\i. x i j) P) = h4/extreal/EXTREAL__SUM__IMAGE (\i. h4/lebesgue/pos__simple__fn__integral m s a (x i)) P
% Assm: h4/lebesgue/pos__simple__fn__integral__sum__alt: !x s m f a P. h4/measure/measure__space m /\ (!i. h4/bool/IN i P ==> h4/measure/pos__simple__fn m (f i) (s i) (a i) (x i)) /\ h4/pred__set/FINITE P /\ ~(P = h4/pred__set/EMPTY) ==> (?c k z. h4/measure/pos__simple__fn m (\t. h4/extreal/EXTREAL__SUM__IMAGE (\i. f i t) P) k c z /\ h4/lebesgue/pos__simple__fn__integral m k c z = h4/extreal/EXTREAL__SUM__IMAGE (\i. h4/lebesgue/pos__simple__fn__integral m (s i) (a i) (x i)) P)
% Assm: h4/lebesgue/psfis__sum: !m f a P. h4/measure/measure__space m /\ (!i. h4/bool/IN i P ==> h4/bool/IN (a i) (h4/lebesgue/psfis m (f i))) /\ h4/pred__set/FINITE P ==> h4/bool/IN (h4/extreal/EXTREAL__SUM__IMAGE a P) (h4/lebesgue/psfis m (\t. h4/extreal/EXTREAL__SUM__IMAGE (\i. f i t) P))
% Assm: h4/lebesgue/lebesgue__monotone__convergence: !m fi f. h4/measure/measure__space m /\ (!i. h4/bool/IN (fi i) (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (fi i x)) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) /\ (!x. h4/extreal/ext__mono__increasing (\i. fi i x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. fi i x) h4/pred__set/UNIV) = f x) ==> h4/lebesgue/pos__fn__integral m f = h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. h4/lebesgue/pos__fn__integral m (fi i)) h4/pred__set/UNIV)
% Assm: h4/lebesgue/integral__sequence: !m f. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) /\ h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> h4/lebesgue/pos__fn__integral m f = h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. h4/lebesgue/pos__fn__integral m (h4/lebesgue/fn__seq m f i)) h4/pred__set/UNIV)
% Assm: h4/lebesgue/measurable__sequence: !m f. h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> (?fi ri. (!x. h4/extreal/ext__mono__increasing (\i. fi i x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. fi i x) h4/pred__set/UNIV) = h4/measure/fn__plus f x) /\ (!i. h4/bool/IN (ri i) (h4/lebesgue/psfis m (fi i))) /\ (!i x. h4/extreal/extreal__le (fi i x) (h4/measure/fn__plus f x)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (fi i x)) /\ h4/lebesgue/pos__fn__integral m (h4/measure/fn__plus f) = h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. h4/lebesgue/pos__fn__integral m (fi i)) h4/pred__set/UNIV)) /\ (?gi vi. (!x. h4/extreal/ext__mono__increasing (\i. gi i x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. gi i x) h4/pred__set/UNIV) = h4/measure/fn__minus f x) /\ (!i. h4/bool/IN (vi i) (h4/lebesgue/psfis m (gi i))) /\ (!i x. h4/extreal/extreal__le (gi i x) (h4/measure/fn__minus f x)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (gi i x)) /\ h4/lebesgue/pos__fn__integral m (h4/measure/fn__minus f) = h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. h4/lebesgue/pos__fn__integral m (gi i)) h4/pred__set/UNIV))
% Assm: h4/lebesgue/pos__fn__integral__add: !m g f. h4/measure/measure__space m /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x) /\ h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (g x)) /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) /\ h4/bool/IN g (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> h4/lebesgue/pos__fn__integral m (\x. h4/extreal/extreal__add (f x) (g x)) = h4/extreal/extreal__add (h4/lebesgue/pos__fn__integral m f) (h4/lebesgue/pos__fn__integral m g)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/TRUTH: T
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/combin/C__DEF: h4/combin/C = (\f x y. f y x)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/pred__set/FINITE__INDUCT: !P. P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ P s ==> (!e. ~h4/bool/IN e s ==> P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> P s)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/extreal/EXTREAL__SUM__IMAGE__THM_c0: !f. h4/extreal/EXTREAL__SUM__IMAGE f h4/pred__set/EMPTY = h4/extreal/extreal__of__num h4/num/0
% Assm: h4/extreal/EXTREAL__SUM__IMAGE__THM_c1: !s f e. h4/pred__set/FINITE s ==> h4/extreal/EXTREAL__SUM__IMAGE f (h4/pred__set/INSERT e s) = h4/extreal/extreal__add (f e) (h4/extreal/EXTREAL__SUM__IMAGE f (h4/pred__set/DELETE s e))
% Assm: h4/pred__set/DELETE__NON__ELEMENT: !x s. ~h4/bool/IN x s <=> h4/pred__set/DELETE s x = s
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/combin/S__DEF: h4/combin/S = (\f g x. f x (g x))
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/extreal/add__rzero: !x. h4/extreal/extreal__add x (h4/extreal/extreal__of__num h4/num/0) = x
% Assm: h4/pred__set/IN__SING: !y x. h4/bool/IN x (h4/pred__set/INSERT y h4/pred__set/EMPTY) <=> x = y
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/lebesgue/psfis__add: !m g f b a. h4/measure/measure__space m /\ h4/bool/IN a (h4/lebesgue/psfis m f) /\ h4/bool/IN b (h4/lebesgue/psfis m g) ==> h4/bool/IN (h4/extreal/extreal__add a b) (h4/lebesgue/psfis m (\x. h4/extreal/extreal__add (f x) (g x)))
% Assm: h4/lebesgue/psfis__zero: !m a. h4/measure/measure__space m ==> (h4/bool/IN a (h4/lebesgue/psfis m (\x. h4/extreal/extreal__of__num h4/num/0)) <=> a = h4/extreal/extreal__of__num h4/num/0)
% Assm: h4/lebesgue/pos__simple__fn__integral__add: !y x s_27 s m g f b a. h4/measure/measure__space m /\ h4/measure/pos__simple__fn m f s a x /\ h4/measure/pos__simple__fn m g s_27 b y ==> (?s_27_27 c z. h4/measure/pos__simple__fn m (\x0. h4/extreal/extreal__add (f x0) (g x0)) s_27_27 c z /\ h4/extreal/extreal__add (h4/lebesgue/pos__simple__fn__integral m s a x) (h4/lebesgue/pos__simple__fn__integral m s_27 b y) = h4/lebesgue/pos__simple__fn__integral m s_27_27 c z)
% Assm: h4/lebesgue/pos__simple__fn__integral__present: !y x s_27 s m g f b a. h4/measure/measure__space m /\ h4/measure/pos__simple__fn m f s a x /\ h4/measure/pos__simple__fn m g s_27 b y ==> (?z z_27 c k. (!t. h4/bool/IN t (h4/measure/m__space m) ==> f t = h4/extreal/EXTREAL__SUM__IMAGE (\i. h4/extreal/extreal__mul (h4/extreal/Normal (z i)) (h4/measure/indicator__fn (c i) t)) k) /\ (!t. h4/bool/IN t (h4/measure/m__space m) ==> g t = h4/extreal/EXTREAL__SUM__IMAGE (\i. h4/extreal/extreal__mul (h4/extreal/Normal (z_27 i)) (h4/measure/indicator__fn (c i) t)) k) /\ h4/lebesgue/pos__simple__fn__integral m s a x = h4/lebesgue/pos__simple__fn__integral m k c z /\ h4/lebesgue/pos__simple__fn__integral m s_27 b y = h4/lebesgue/pos__simple__fn__integral m k c z_27 /\ h4/pred__set/FINITE k /\ (!i. h4/bool/IN i k ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (z i)) /\ (!i. h4/bool/IN i k ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (z_27 i)) /\ (!i j. h4/bool/IN i k /\ h4/bool/IN j k /\ ~(i = j) ==> h4/pred__set/DISJOINT (c i) (c j)) /\ (!i. h4/bool/IN i k ==> h4/bool/IN (c i) (h4/measure/measurable__sets m)) /\ h4/pred__set/BIGUNION (h4/pred__set/IMAGE c k) = h4/measure/m__space m)
% Assm: h4/pred__set/EMPTY__DELETE: !x. h4/pred__set/DELETE h4/pred__set/EMPTY x = h4/pred__set/EMPTY
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/lebesgue/lemma__fn__mono__increasing: !x m f. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x) ==> h4/extreal/ext__mono__increasing (\n. h4/lebesgue/fn__seq m f n x)
% Assm: h4/lebesgue/lemma__fn__sup: !x m f. h4/bool/IN x (h4/measure/m__space m) /\ h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (\n. h4/lebesgue/fn__seq m f n x) h4/pred__set/UNIV) = f x
% Assm: h4/lebesgue/lemma__fn__5: !x n m f. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x) ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/lebesgue/fn__seq m f n x)
% Assm: h4/lebesgue/lemma__fn__in__psfis: !n m f. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) /\ h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> h4/bool/IN (h4/lebesgue/fn__seq__integral m f n) (h4/lebesgue/psfis m (h4/lebesgue/fn__seq m f n))
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/lebesgue/pos__simple__fn__integral__add__alt: !y x s m g f a. h4/measure/measure__space m /\ h4/measure/pos__simple__fn m f s a x /\ h4/measure/pos__simple__fn m g s a y ==> h4/extreal/extreal__add (h4/lebesgue/pos__simple__fn__integral m s a x) (h4/lebesgue/pos__simple__fn__integral m s a y) = h4/lebesgue/pos__simple__fn__integral m s a (\i. h4/realax/real__add (x i) (y i))
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/measure/pos__simple__fn__def: !x s m f a. h4/measure/pos__simple__fn m f s a x <=> (!t. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f t)) /\ (!t. h4/bool/IN t (h4/measure/m__space m) ==> f t = h4/extreal/EXTREAL__SUM__IMAGE (\i. h4/extreal/extreal__mul (h4/extreal/Normal (x i)) (h4/measure/indicator__fn (a i) t)) s) /\ (!i. h4/bool/IN i s ==> h4/bool/IN (a i) (h4/measure/measurable__sets m)) /\ h4/pred__set/FINITE s /\ (!i. h4/bool/IN i s ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (x i)) /\ (!i j. h4/bool/IN i s /\ h4/bool/IN j s /\ ~(i = j) ==> h4/pred__set/DISJOINT (a i) (a j)) /\ h4/pred__set/BIGUNION (h4/pred__set/IMAGE a s) = h4/measure/m__space m
% Assm: h4/extreal/EXTREAL__SUM__IMAGE__0: !s f. h4/pred__set/FINITE s /\ (!x. h4/bool/IN x s ==> f x = h4/extreal/extreal__of__num h4/num/0) ==> h4/extreal/EXTREAL__SUM__IMAGE f s = h4/extreal/extreal__of__num h4/num/0
% Assm: h4/extreal/le__refl: !x. h4/extreal/extreal__le x x
% Assm: h4/extreal/mul__lzero: !x. h4/extreal/extreal__mul (h4/extreal/extreal__of__num h4/num/0) x = h4/extreal/extreal__of__num h4/num/0
% Assm: h4/real__sigma/REAL__SUM__IMAGE__THM_c0: !f. h4/real__sigma/REAL__SUM__IMAGE f h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0
% Assm: h4/lebesgue/pos__simple__fn__add__alt: !y x s m g f a. h4/measure/measure__space m /\ h4/measure/pos__simple__fn m f s a x /\ h4/measure/pos__simple__fn m g s a y ==> h4/measure/pos__simple__fn m (\t. h4/extreal/extreal__add (f t) (g t)) s a (\i. h4/realax/real__add (x i) (y i))
% Assm: h4/real__sigma/REAL__SUM__IMAGE__THM_c1: !s f e. h4/pred__set/FINITE s ==> h4/real__sigma/REAL__SUM__IMAGE f (h4/pred__set/INSERT e s) = h4/realax/real__add (f e) (h4/real__sigma/REAL__SUM__IMAGE f (h4/pred__set/DELETE s e))
% Assm: h4/extreal/extreal__of__num__def: !n. h4/extreal/extreal__of__num n = h4/extreal/Normal (h4/real/real__of__num n)
% Assm: h4/real/le__int_c0: !n m. h4/real/real__lte (h4/real/real__of__num n) (h4/real/real__of__num m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/real/REAL__ADD__RID: !x. h4/realax/real__add x (h4/real/real__of__num h4/num/0) = x
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/lebesgue/IN__psfis__eq: !r m f. h4/bool/IN r (h4/lebesgue/psfis m f) <=> (?s a x. h4/measure/pos__simple__fn m f s a x /\ r = h4/lebesgue/pos__simple__fn__integral m s a x)
% Assm: h4/measure/IN__MEASURABLE__BOREL__POS__SIMPLE__FN: !m f. h4/measure/measure__space m /\ (?s a x. h4/measure/pos__simple__fn m f s a x) ==> h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/lebesgue/lemma__fn__upper__bounded: !x n m f. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x) ==> h4/extreal/extreal__le (h4/lebesgue/fn__seq m f n x) (f x)
% Assm: h4/measure/IN__MEASURABLE__BOREL__FN__MINUS: !f a. h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) ==> h4/bool/IN (h4/measure/fn__minus f) (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/measure/IN__MEASURABLE__BOREL__FN__PLUS: !f a. h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) ==> h4/bool/IN (h4/measure/fn__plus f) (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/measure/FN__MINUS__POS: !x g. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/measure/fn__minus g x)
% Assm: h4/measure/FN__PLUS__POS: !x g. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/measure/fn__plus g x)
% Assm: h4/lebesgue/lebesgue__monotone__convergence__lemma: !r_27 m g fi f. h4/measure/measure__space m /\ (!i. h4/bool/IN (fi i) (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)) /\ (!x. h4/extreal/ext__mono__increasing (\i. fi i x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. fi i x) h4/pred__set/UNIV) = f x) /\ h4/bool/IN r_27 (h4/lebesgue/psfis m g) /\ (!x. h4/extreal/extreal__le (g x) (f x)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (fi i x)) ==> h4/extreal/extreal__le r_27 (h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. h4/lebesgue/pos__fn__integral m (fi i)) h4/pred__set/UNIV))
% Assm: h4/lebesgue/pos__fn__integral__mono__mspace: !m g f. h4/measure/measure__space m /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__le (g x) (f x)) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (g x)) ==> h4/extreal/extreal__le (h4/lebesgue/pos__fn__integral m g) (h4/lebesgue/pos__fn__integral m f)
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/lebesgue/pos__fn__integral__def: !m f. h4/lebesgue/pos__fn__integral m f = h4/extreal/extreal__sup (h4/pred__set/GSPEC (\r. h4/pair/_2C r (?g. h4/bool/IN r (h4/lebesgue/psfis m g) /\ (!x. h4/extreal/extreal__le (g x) (f x)))))
% Assm: h4/extreal/sup__le: !x p. h4/extreal/extreal__le (h4/extreal/extreal__sup p) x <=> (!y. p y ==> h4/extreal/extreal__le y x)
% Assm: h4/extreal/le__antisym: !y x. h4/extreal/extreal__le x y /\ h4/extreal/extreal__le y x <=> x = y
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/pred__set/IN__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = f x /\ h4/bool/IN x s)
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/lebesgue/pos__fn__integral__pos: !m f. h4/measure/measure__space m /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/lebesgue/pos__fn__integral m f)
% Assm: h4/lebesgue/pos__fn__integral__pos__simple__fn: !x s m f a. h4/measure/measure__space m /\ h4/measure/pos__simple__fn m f s a x ==> h4/lebesgue/pos__fn__integral m f = h4/lebesgue/pos__simple__fn__integral m s a x
% Assm: h4/lebesgue/pos__fn__integral__mono: !m g f. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x) /\ h4/extreal/extreal__le (f x) (g x)) ==> h4/extreal/extreal__le (h4/lebesgue/pos__fn__integral m f) (h4/lebesgue/pos__fn__integral m g)
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/measure/FN__PLUS__POS__ID: !g. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (g x)) ==> h4/measure/fn__plus g = g
% Assm: h4/extreal/sup__add__mono: !g f. (!n. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f n)) /\ (!n. h4/extreal/extreal__le (f n) (f (h4/num/SUC n))) /\ (!n. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (g n)) /\ (!n. h4/extreal/extreal__le (g n) (g (h4/num/SUC n))) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (\n. h4/extreal/extreal__add (f n) (g n)) h4/pred__set/UNIV) = h4/extreal/extreal__add (h4/extreal/extreal__sup (h4/pred__set/IMAGE f h4/pred__set/UNIV)) (h4/extreal/extreal__sup (h4/pred__set/IMAGE g h4/pred__set/UNIV))
% Assm: h4/extreal/ext__mono__increasing__def: !f. h4/extreal/ext__mono__increasing f <=> (!m n. h4/arithmetic/_3C_3D m n ==> h4/extreal/extreal__le (f m) (f n))
% Assm: h4/extreal/le__add2: !z y x w. h4/extreal/extreal__le w x /\ h4/extreal/extreal__le y z ==> h4/extreal/extreal__le (h4/extreal/extreal__add w y) (h4/extreal/extreal__add x z)
% Assm: h4/extreal/le__add: !y x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) x /\ h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) y ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/extreal/extreal__add x y)
% Assm: h4/extreal/ext__mono__increasing__suc: !f. h4/extreal/ext__mono__increasing f <=> (!n. h4/extreal/extreal__le (f n) (f (h4/num/SUC n)))
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Goal: !s m f. h4/pred__set/FINITE s /\ h4/measure/measure__space m /\ (!i. h4/bool/IN i s ==> (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f i x))) /\ (!i. h4/bool/IN i s ==> h4/bool/IN (f i) (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)) ==> h4/lebesgue/pos__fn__integral m (\x. h4/extreal/EXTREAL__SUM__IMAGE (\i. f i x) s) = h4/extreal/EXTREAL__SUM__IMAGE (\i. h4/lebesgue/pos__fn__integral m (f i)) s
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_sum]: !_4. (!m s a x i. happ (happ (happ (happ (happ _4 m) s) a) x) i = h4/lebesgue/pos__simple__fn__integral m s a (happ x i)) ==> (!_3. (!x i j. happ (happ (happ _3 x) i) j = happ (happ x j) i) ==> (!_2. (!x P i. happ (happ (happ _2 x) P) i = h4/real__sigma/REAL__SUM__IMAGE (happ (happ _3 x) i) P) ==> (!_1. (!f t i. happ (happ (happ _1 f) t) i = happ (happ f i) t) ==> (!_0. (!f P t. happ (happ (happ _0 f) P) t = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ _1 f) t) P) ==> (!x s m f a P. h4/measure/measure__space m /\ (!i. h4/bool/IN i P ==> h4/measure/pos__simple__fn m (happ f i) s a (happ x i)) /\ h4/pred__set/FINITE P /\ ~(P = h4/pred__set/EMPTY) ==> h4/measure/pos__simple__fn m (happ (happ _0 f) P) s a (happ (happ _2 x) P) /\ h4/lebesgue/pos__simple__fn__integral m s a (happ (happ _2 x) P) = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ (happ (happ _4 m) s) a) x) P)))))
% Assm [h4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_sumu_u_alt]: !_2. (!m s a x i. happ (happ (happ (happ (happ _2 m) s) a) x) i = h4/lebesgue/pos__simple__fn__integral m (happ s i) (happ a i) (happ x i)) ==> (!_1. (!f t i. happ (happ (happ _1 f) t) i = happ (happ f i) t) ==> (!_0. (!f P t. happ (happ (happ _0 f) P) t = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ _1 f) t) P) ==> (!x s m f a P. h4/measure/measure__space m /\ (!i. h4/bool/IN i P ==> h4/measure/pos__simple__fn m (happ f i) (happ s i) (happ a i) (happ x i)) /\ h4/pred__set/FINITE P /\ ~(P = h4/pred__set/EMPTY) ==> (?c k z. h4/measure/pos__simple__fn m (happ (happ _0 f) P) k c z /\ h4/lebesgue/pos__simple__fn__integral m k c z = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ (happ (happ _2 m) s) a) x) P))))
% Assm [h4s_lebesgues_psfisu_u_sum]: !_1. (!f t i. happ (happ (happ _1 f) t) i = happ (happ f i) t) ==> (!_0. (!f P t. happ (happ (happ _0 f) P) t = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ _1 f) t) P) ==> (!m f a P. h4/measure/measure__space m /\ (!i. h4/bool/IN i P ==> h4/bool/IN (happ a i) (h4/lebesgue/psfis m (happ f i))) /\ h4/pred__set/FINITE P ==> h4/bool/IN (h4/extreal/EXTREAL__SUM__IMAGE a P) (h4/lebesgue/psfis m (happ (happ _0 f) P))))
% Assm [h4s_lebesgues_lebesgueu_u_monotoneu_u_convergence]: !_1. (!m fi i. happ (happ (happ _1 m) fi) i = h4/lebesgue/pos__fn__integral m (happ fi i)) ==> (!_0. (!fi x i. happ (happ (happ _0 fi) x) i = happ (happ fi i) x) ==> (!m fi f. h4/measure/measure__space m /\ (!i. h4/bool/IN (happ fi i) (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (happ fi i) x)) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) /\ (!x. h4/extreal/ext__mono__increasing (happ (happ _0 fi) x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 fi) x) h4/pred__set/UNIV) = happ f x) ==> h4/lebesgue/pos__fn__integral m f = h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _1 m) fi) h4/pred__set/UNIV)))
% Assm [h4s_lebesgues_integralu_u_sequence]: !_0. (!m f i. happ (happ (happ _0 m) f) i = h4/lebesgue/pos__fn__integral m (h4/lebesgue/fn__seq m f i)) ==> (!m f. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) /\ h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> h4/lebesgue/pos__fn__integral m f = h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 m) f) h4/pred__set/UNIV))
% Assm [h4s_lebesgues_measurableu_u_sequence]: !_1. (!m fi i. happ (happ (happ _1 m) fi) i = h4/lebesgue/pos__fn__integral m (happ fi i)) ==> (!_0. (!fi x i. happ (happ (happ _0 fi) x) i = happ (happ fi i) x) ==> (!m f. h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> (?fi ri. (!x. h4/extreal/ext__mono__increasing (happ (happ _0 fi) x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 fi) x) h4/pred__set/UNIV) = happ (h4/measure/fn__plus f) x) /\ (!i. h4/bool/IN (happ ri i) (h4/lebesgue/psfis m (happ fi i))) /\ (!i x. h4/extreal/extreal__le (happ (happ fi i) x) (happ (h4/measure/fn__plus f) x)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (happ fi i) x)) /\ h4/lebesgue/pos__fn__integral m (h4/measure/fn__plus f) = h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _1 m) fi) h4/pred__set/UNIV)) /\ (?gi vi. (!x. h4/extreal/ext__mono__increasing (happ (happ _0 gi) x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 gi) x) h4/pred__set/UNIV) = happ (h4/measure/fn__minus f) x) /\ (!i. h4/bool/IN (happ vi i) (h4/lebesgue/psfis m (happ gi i))) /\ (!i x. h4/extreal/extreal__le (happ (happ gi i) x) (happ (h4/measure/fn__minus f) x)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (happ gi i) x)) /\ h4/lebesgue/pos__fn__integral m (h4/measure/fn__minus f) = h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _1 m) gi) h4/pred__set/UNIV))))
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_add]: !_0. (!f g x. happ (happ (happ _0 f) g) x = h4/extreal/extreal__add (happ f x) (happ g x)) ==> (!m g f. h4/measure/measure__space m /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x) /\ h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ g x)) /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) /\ h4/bool/IN g (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> h4/lebesgue/pos__fn__integral m (happ (happ _0 f) g) = h4/extreal/extreal__add (h4/lebesgue/pos__fn__integral m f) (h4/lebesgue/pos__fn__integral m g))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_combins_Iu_u_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_combins_Cu_u_DEF]: !x x x. h4/combin/C x x x = happ (happ x x) x
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_predu_u_sets_FINITEu_u_INDUCT]: !P. happ P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ happ P s ==> (!e. ~h4/bool/IN e s ==> happ P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> happ P s)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_THMu_c0]: !f. h4/extreal/EXTREAL__SUM__IMAGE f h4/pred__set/EMPTY = h4/extreal/extreal__of__num h4/num/0
% Assm [h4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_THMu_c1]: !s f e. h4/pred__set/FINITE s ==> h4/extreal/EXTREAL__SUM__IMAGE f (h4/pred__set/INSERT e s) = h4/extreal/extreal__add (happ f e) (h4/extreal/EXTREAL__SUM__IMAGE f (h4/pred__set/DELETE s e))
% Assm [h4s_predu_u_sets_DELETEu_u_NONu_u_ELEMENT]: !x s. ~h4/bool/IN x s <=> h4/pred__set/DELETE s x = s
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_combins_Su_u_DEF]: !x x x. h4/combin/S x x x = happ (happ x x) (happ x x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_extreals_addu_u_rzero]: !x. h4/extreal/extreal__add x (h4/extreal/extreal__of__num h4/num/0) = x
% Assm [h4s_predu_u_sets_INu_u_SING]: !y x. h4/bool/IN x (h4/pred__set/INSERT y h4/pred__set/EMPTY) <=> x = y
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_lebesgues_psfisu_u_add]: !_0. (!f g x. happ (happ (happ _0 f) g) x = h4/extreal/extreal__add (happ f x) (happ g x)) ==> (!m g f b a. h4/measure/measure__space m /\ h4/bool/IN a (h4/lebesgue/psfis m f) /\ h4/bool/IN b (h4/lebesgue/psfis m g) ==> h4/bool/IN (h4/extreal/extreal__add a b) (h4/lebesgue/psfis m (happ (happ _0 f) g)))
% Assm [h4s_lebesgues_psfisu_u_zero]: !_0. (!x. happ _0 x = h4/extreal/extreal__of__num h4/num/0) ==> (!m a. h4/measure/measure__space m ==> (h4/bool/IN a (h4/lebesgue/psfis m _0) <=> a = h4/extreal/extreal__of__num h4/num/0))
% Assm [h4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_add]: !_0. (!f g x0. happ (happ (happ _0 f) g) x0 = h4/extreal/extreal__add (happ f x0) (happ g x0)) ==> (!y x s_27 s m g f b a. h4/measure/measure__space m /\ h4/measure/pos__simple__fn m f s a x /\ h4/measure/pos__simple__fn m g s_27 b y ==> (?s_27_27 c z. h4/measure/pos__simple__fn m (happ (happ _0 f) g) s_27_27 c z /\ h4/extreal/extreal__add (h4/lebesgue/pos__simple__fn__integral m s a x) (h4/lebesgue/pos__simple__fn__integral m s_27 b y) = h4/lebesgue/pos__simple__fn__integral m s_27_27 c z))
% Assm [h4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_present]: !_0. (!z c t i. happ (happ (happ (happ _0 z) c) t) i = h4/extreal/extreal__mul (h4/extreal/Normal (happ z i)) (h4/measure/indicator__fn (happ c i) t)) ==> (!y x s_27 s m g f b a. h4/measure/measure__space m /\ h4/measure/pos__simple__fn m f s a x /\ h4/measure/pos__simple__fn m g s_27 b y ==> (?z z_27 c k. (!t. h4/bool/IN t (h4/measure/m__space m) ==> happ f t = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ (happ _0 z) c) t) k) /\ (!t. h4/bool/IN t (h4/measure/m__space m) ==> happ g t = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ (happ _0 z_27) c) t) k) /\ h4/lebesgue/pos__simple__fn__integral m s a x = h4/lebesgue/pos__simple__fn__integral m k c z /\ h4/lebesgue/pos__simple__fn__integral m s_27 b y = h4/lebesgue/pos__simple__fn__integral m k c z_27 /\ h4/pred__set/FINITE k /\ (!i. h4/bool/IN i k ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ z i)) /\ (!i. h4/bool/IN i k ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ z_27 i)) /\ (!i j. h4/bool/IN i k /\ h4/bool/IN j k /\ ~(i = j) ==> h4/pred__set/DISJOINT (happ c i) (happ c j)) /\ (!i. h4/bool/IN i k ==> h4/bool/IN (happ c i) (h4/measure/measurable__sets m)) /\ h4/pred__set/BIGUNION (h4/pred__set/IMAGE c k) = h4/measure/m__space m))
% Assm [h4s_predu_u_sets_EMPTYu_u_DELETE]: !x. h4/pred__set/DELETE h4/pred__set/EMPTY x = h4/pred__set/EMPTY
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_lebesgues_lemmau_u_fnu_u_monou_u_increasing]: !_0. (!m f x n. happ (happ (happ (happ _0 m) f) x) n = happ (h4/lebesgue/fn__seq m f n) x) ==> (!x m f. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x) ==> h4/extreal/ext__mono__increasing (happ (happ (happ _0 m) f) x))
% Assm [h4s_lebesgues_lemmau_u_fnu_u_sup]: !_0. (!m f x n. happ (happ (happ (happ _0 m) f) x) n = happ (h4/lebesgue/fn__seq m f n) x) ==> (!x m f. h4/bool/IN x (h4/measure/m__space m) /\ h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ (happ _0 m) f) x) h4/pred__set/UNIV) = happ f x)
% Assm [h4s_lebesgues_lemmau_u_fnu_u_5]: !x n m f. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x) ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (h4/lebesgue/fn__seq m f n) x)
% Assm [h4s_lebesgues_lemmau_u_fnu_u_inu_u_psfis]: !n m f. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) /\ h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> h4/bool/IN (h4/lebesgue/fn__seq__integral m f n) (h4/lebesgue/psfis m (h4/lebesgue/fn__seq m f n))
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_addu_u_alt]: !_0. (!x y i. happ (happ (happ _0 x) y) i = h4/realax/real__add (happ x i) (happ y i)) ==> (!y x s m g f a. h4/measure/measure__space m /\ h4/measure/pos__simple__fn m f s a x /\ h4/measure/pos__simple__fn m g s a y ==> h4/extreal/extreal__add (h4/lebesgue/pos__simple__fn__integral m s a x) (h4/lebesgue/pos__simple__fn__integral m s a y) = h4/lebesgue/pos__simple__fn__integral m s a (happ (happ _0 x) y))
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_measures_posu_u_simpleu_u_fnu_u_def]: !_0. (!x a t i. happ (happ (happ (happ _0 x) a) t) i = h4/extreal/extreal__mul (h4/extreal/Normal (happ x i)) (h4/measure/indicator__fn (happ a i) t)) ==> (!x s m f a. h4/measure/pos__simple__fn m f s a x <=> (!t. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f t)) /\ (!t. h4/bool/IN t (h4/measure/m__space m) ==> happ f t = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ (happ _0 x) a) t) s) /\ (!i. h4/bool/IN i s ==> h4/bool/IN (happ a i) (h4/measure/measurable__sets m)) /\ h4/pred__set/FINITE s /\ (!i. h4/bool/IN i s ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ x i)) /\ (!i j. h4/bool/IN i s /\ h4/bool/IN j s /\ ~(i = j) ==> h4/pred__set/DISJOINT (happ a i) (happ a j)) /\ h4/pred__set/BIGUNION (h4/pred__set/IMAGE a s) = h4/measure/m__space m)
% Assm [h4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_0]: !s f. h4/pred__set/FINITE s /\ (!x. h4/bool/IN x s ==> happ f x = h4/extreal/extreal__of__num h4/num/0) ==> h4/extreal/EXTREAL__SUM__IMAGE f s = h4/extreal/extreal__of__num h4/num/0
% Assm [h4s_extreals_leu_u_refl]: !x. h4/extreal/extreal__le x x
% Assm [h4s_extreals_mulu_u_lzero]: !x. h4/extreal/extreal__mul (h4/extreal/extreal__of__num h4/num/0) x = h4/extreal/extreal__of__num h4/num/0
% Assm [h4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_THMu_c0]: !f. h4/real__sigma/REAL__SUM__IMAGE f h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0
% Assm [h4s_lebesgues_posu_u_simpleu_u_fnu_u_addu_u_alt]: !_1. (!x y i. happ (happ (happ _1 x) y) i = h4/realax/real__add (happ x i) (happ y i)) ==> (!_0. (!f g t. happ (happ (happ _0 f) g) t = h4/extreal/extreal__add (happ f t) (happ g t)) ==> (!y x s m g f a. h4/measure/measure__space m /\ h4/measure/pos__simple__fn m f s a x /\ h4/measure/pos__simple__fn m g s a y ==> h4/measure/pos__simple__fn m (happ (happ _0 f) g) s a (happ (happ _1 x) y)))
% Assm [h4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_THMu_c1]: !s f e. h4/pred__set/FINITE s ==> h4/real__sigma/REAL__SUM__IMAGE f (h4/pred__set/INSERT e s) = h4/realax/real__add (happ f e) (h4/real__sigma/REAL__SUM__IMAGE f (h4/pred__set/DELETE s e))
% Assm [h4s_extreals_extrealu_u_ofu_u_numu_u_def]: !n. h4/extreal/extreal__of__num n = h4/extreal/Normal (h4/real/real__of__num n)
% Assm [h4s_reals_leu_u_intu_c0]: !n m. h4/real/real__lte (h4/real/real__of__num n) (h4/real/real__of__num m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_reals_REALu_u_ADDu_u_RID]: !x. h4/realax/real__add x (h4/real/real__of__num h4/num/0) = x
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_lebesgues_INu_u_psfisu_u_eq]: !r m f. h4/bool/IN r (h4/lebesgue/psfis m f) <=> (?s a x. h4/measure/pos__simple__fn m f s a x /\ r = h4/lebesgue/pos__simple__fn__integral m s a x)
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_POSu_u_SIMPLEu_u_FN]: !m f. h4/measure/measure__space m /\ (?s a x. h4/measure/pos__simple__fn m f s a x) ==> h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)
% Assm [h4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_lebesgues_lemmau_u_fnu_u_upperu_u_bounded]: !x n m f. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x) ==> h4/extreal/extreal__le (happ (h4/lebesgue/fn__seq m f n) x) (happ f x)
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_FNu_u_MINUS]: !f a. h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) ==> h4/bool/IN (h4/measure/fn__minus f) (h4/measure/measurable a h4/measure/Borel)
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_FNu_u_PLUS]: !f a. h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) ==> h4/bool/IN (h4/measure/fn__plus f) (h4/measure/measurable a h4/measure/Borel)
% Assm [h4s_measures_FNu_u_MINUSu_u_POS]: !x g. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (h4/measure/fn__minus g) x)
% Assm [h4s_measures_FNu_u_PLUSu_u_POS]: !x g. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (h4/measure/fn__plus g) x)
% Assm [h4s_lebesgues_lebesgueu_u_monotoneu_u_convergenceu_u_lemma]: !_1. (!m fi i. happ (happ (happ _1 m) fi) i = h4/lebesgue/pos__fn__integral m (happ fi i)) ==> (!_0. (!fi x i. happ (happ (happ _0 fi) x) i = happ (happ fi i) x) ==> (!r_27 m g fi f. h4/measure/measure__space m /\ (!i. h4/bool/IN (happ fi i) (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)) /\ (!x. h4/extreal/ext__mono__increasing (happ (happ _0 fi) x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 fi) x) h4/pred__set/UNIV) = happ f x) /\ h4/bool/IN r_27 (h4/lebesgue/psfis m g) /\ (!x. h4/extreal/extreal__le (happ g x) (happ f x)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (happ fi i) x)) ==> h4/extreal/extreal__le r_27 (h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _1 m) fi) h4/pred__set/UNIV))))
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_monou_u_mspace]: !m g f. h4/measure/measure__space m /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__le (happ g x) (happ f x)) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ g x)) ==> h4/extreal/extreal__le (h4/lebesgue/pos__fn__integral m g) (h4/lebesgue/pos__fn__integral m f)
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_def]: !_0. (!m f r. ?v. (v <=> (?g. h4/bool/IN r (h4/lebesgue/psfis m g) /\ (!x. h4/extreal/extreal__le (happ g x) (happ f x)))) /\ happ (happ (happ _0 m) f) r = h4/pair/_2C r v) ==> (!m f. h4/lebesgue/pos__fn__integral m f = h4/extreal/extreal__sup (h4/pred__set/GSPEC (happ (happ _0 m) f)))
% Assm [h4s_extreals_supu_u_le]: !x p. h4/extreal/extreal__le (h4/extreal/extreal__sup p) x <=> (!y. happ p y ==> h4/extreal/extreal__le y x)
% Assm [h4s_extreals_leu_u_antisym]: !y x. h4/extreal/extreal__le x y /\ h4/extreal/extreal__le y x <=> x = y
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_predu_u_sets_INu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = happ f x /\ h4/bool/IN x s)
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_markers_Abbrevu_u_def]: !x. h4/marker/Abbrev x <=> x
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_pos]: !m f. h4/measure/measure__space m /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/lebesgue/pos__fn__integral m f)
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_posu_u_simpleu_u_fn]: !x s m f a. h4/measure/measure__space m /\ h4/measure/pos__simple__fn m f s a x ==> h4/lebesgue/pos__fn__integral m f = h4/lebesgue/pos__simple__fn__integral m s a x
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_mono]: !m g f. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x) /\ h4/extreal/extreal__le (happ f x) (happ g x)) ==> h4/extreal/extreal__le (h4/lebesgue/pos__fn__integral m f) (h4/lebesgue/pos__fn__integral m g)
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_measures_FNu_u_PLUSu_u_POSu_u_ID]: !g. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ g x)) ==> h4/measure/fn__plus g = g
% Assm [h4s_extreals_supu_u_addu_u_mono]: !_0. (!f g n. happ (happ (happ _0 f) g) n = h4/extreal/extreal__add (happ f n) (happ g n)) ==> (!g f. (!n. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f n)) /\ (!n. h4/extreal/extreal__le (happ f n) (happ f (h4/num/SUC n))) /\ (!n. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ g n)) /\ (!n. h4/extreal/extreal__le (happ g n) (happ g (h4/num/SUC n))) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 f) g) h4/pred__set/UNIV) = h4/extreal/extreal__add (h4/extreal/extreal__sup (h4/pred__set/IMAGE f h4/pred__set/UNIV)) (h4/extreal/extreal__sup (h4/pred__set/IMAGE g h4/pred__set/UNIV)))
% Assm [h4s_extreals_extu_u_monou_u_increasingu_u_def]: !f. h4/extreal/ext__mono__increasing f <=> (!m n. h4/arithmetic/_3C_3D m n ==> h4/extreal/extreal__le (happ f m) (happ f n))
% Assm [h4s_extreals_leu_u_add2]: !z y x w. h4/extreal/extreal__le w x /\ h4/extreal/extreal__le y z ==> h4/extreal/extreal__le (h4/extreal/extreal__add w y) (h4/extreal/extreal__add x z)
% Assm [h4s_extreals_leu_u_add]: !y x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) x /\ h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) y ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/extreal/extreal__add x y)
% Assm [h4s_extreals_extu_u_monou_u_increasingu_u_suc]: !f. h4/extreal/ext__mono__increasing f <=> (!n. h4/extreal/extreal__le (happ f n) (happ f (h4/num/SUC n)))
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Goal: !_2. (!m f i. happ (happ (happ _2 m) f) i = h4/lebesgue/pos__fn__integral m (happ f i)) ==> (!_1. (!f x i. happ (happ (happ _1 f) x) i = happ (happ f i) x) ==> (!_0. (!f s x. happ (happ (happ _0 f) s) x = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ _1 f) x) s) ==> (!s m f. h4/pred__set/FINITE s /\ h4/measure/measure__space m /\ (!i. h4/bool/IN i s ==> (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (happ f i) x))) /\ (!i. h4/bool/IN i s ==> h4/bool/IN (happ f i) (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)) ==> h4/lebesgue/pos__fn__integral m (happ (happ _0 f) s) = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ _2 m) f) s)))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q1361807,TV_Q1361803]: ![V_f, V_g]: (![V_x]: s(TV_Q1361803,happ(s(t_fun(TV_Q1361807,TV_Q1361803),V_f),s(TV_Q1361807,V_x))) = s(TV_Q1361803,happ(s(t_fun(TV_Q1361807,TV_Q1361803),V_g),s(TV_Q1361807,V_x))) => s(t_fun(TV_Q1361807,TV_Q1361803),V_f) = s(t_fun(TV_Q1361807,TV_Q1361803),V_g))).
fof(ah4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_sum, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_4]: (![V_m, V_s, V_a, V_x, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal)))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal))))),V_uu_4),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_bool),V_s))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a))),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x))),s(TV_u_27b,V_i))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x),s(TV_u_27b,V_i))))) => ![V_uu_3]: (![V_x, V_i, V_j]: s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27b,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_realaxs_real))),V_uu_3),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x))),s(t_h4s_nums_num,V_i))),s(TV_u_27b,V_j))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x),s(TV_u_27b,V_j))),s(t_h4s_nums_num,V_i))) => ![V_uu_2]: (![V_x, V_P, V_i]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_2),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x))),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_h4s_nums_num,V_i))) = s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27b,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_realaxs_real))),V_uu_3),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x))),s(t_h4s_nums_num,V_i))),s(t_fun(TV_u_27b,t_bool),V_P))) => ![V_uu_1]: (![V_f, V_t, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(TV_u_27a,V_t))),s(TV_u_27b,V_i))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(TV_u_27b,V_i))),s(TV_u_27a,V_t))) => ![V_uu_0]: (![V_f, V_P, V_t]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27a,V_t))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(TV_u_27a,V_t))),s(t_fun(TV_u_27b,t_bool),V_P))) => ![V_x, V_s, V_m, V_f, V_a, V_P]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (![V_i]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) => p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(TV_u_27b,V_i))),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x),s(TV_u_27b,V_i))))))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),V_P)))) & ~ (s(t_fun(TV_u_27b,t_bool),V_P) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty))))) => (p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_2),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x))),s(t_fun(TV_u_27b,t_bool),V_P)))))) & s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_2),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x))),s(t_fun(TV_u_27b,t_bool),V_P))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal)))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal))))),V_uu_4),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_bool),V_s))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a))),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x))),s(t_fun(TV_u_27b,t_bool),V_P))))))))))).
fof(ah4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_sumu_u_alt, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_2]: (![V_m, V_s, V_a, V_x, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal)))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal))))),V_uu_2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_bool)),V_s))),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),V_a))),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x))),s(TV_u_27b,V_i))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_bool)),V_s),s(TV_u_27b,V_i))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),V_a),s(TV_u_27b,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x),s(TV_u_27b,V_i))))) => ![V_uu_1]: (![V_f, V_t, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(TV_u_27a,V_t))),s(TV_u_27b,V_i))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(TV_u_27b,V_i))),s(TV_u_27a,V_t))) => ![V_uu_0]: (![V_f, V_P, V_t]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27a,V_t))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(TV_u_27a,V_t))),s(t_fun(TV_u_27b,t_bool),V_P))) => ![V_x, V_s, V_m, V_f, V_a, V_P]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (![V_i]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) => p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(TV_u_27b,V_i))),s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_bool)),V_s),s(TV_u_27b,V_i))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),V_a),s(TV_u_27b,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x),s(TV_u_27b,V_i))))))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),V_P)))) & ~ (s(t_fun(TV_u_27b,t_bool),V_P) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty))))) => ?[V_c, V_k, V_z]: (p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(t_h4s_nums_num,t_bool),V_k),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_z)))) & s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_k),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_z))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal)))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),t_fun(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_fun(TV_u_27b,t_h4s_extreals_extreal))))),V_uu_2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_bool)),V_s))),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),V_a))),s(t_fun(TV_u_27b,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_x))),s(t_fun(TV_u_27b,t_bool),V_P))))))))).
fof(ah4s_lebesgues_psfisu_u_sum, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_f, V_t, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(TV_u_27a,V_t))),s(TV_u_27b,V_i))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(TV_u_27b,V_i))),s(TV_u_27a,V_t))) => ![V_uu_0]: (![V_f, V_P, V_t]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(t_fun(TV_u_27b,t_bool),V_P))),s(TV_u_27a,V_t))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(TV_u_27a,V_t))),s(t_fun(TV_u_27b,t_bool),V_P))) => ![V_m, V_f, V_a, V_P]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (![V_i]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) => p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),V_a),s(TV_u_27b,V_i))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(TV_u_27b,V_i))))))))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),V_P)))))) => p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),V_a),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(t_fun(TV_u_27b,t_bool),V_P)))))))))))).
fof(ah4s_lebesgues_lebesgueu_u_monotoneu_u_convergence, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_m, V_fi, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))))) => ![V_uu_0]: (![V_fi, V_x, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))) => ![V_m, V_fi, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (![V_i]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))) & (![V_i, V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x)))))) & (![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))) & (![V_x]: p(s(t_bool,h4s_extreals_extu_u_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x)))))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))))))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))))))).
fof(ah4s_lebesgues_integralu_u_sequence, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_m, V_f, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_lebesgues_fnu_u_seq(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_i))))) => ![V_m, V_f]: ((![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))) & (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ)))))))).
fof(ah4s_lebesgues_measurableu_u_sequence, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_m, V_fi, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))))) => ![V_uu_0]: (![V_fi, V_x, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))) => ![V_m, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel))))))) => (?[V_fi, V_ri]: (![V_x]: p(s(t_bool,h4s_extreals_extu_u_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x)))))) & (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x)))) & (![V_i]: p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_ri),s(t_h4s_nums_num,V_i))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i)))))))) & (![V_i, V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x)))))) & (![V_i, V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x)))))) & s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ)))))))))) & ?[V_gi, V_vi]: (![V_x]: p(s(t_bool,h4s_extreals_extu_u_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_gi))),s(TV_u_27a,V_x)))))) & (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_gi))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_minus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x)))) & (![V_i]: p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_vi),s(t_h4s_nums_num,V_i))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_gi),s(t_h4s_nums_num,V_i)))))))) & (![V_i, V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_gi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_minus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x)))))) & (![V_i, V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_gi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x)))))) & s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_minus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_gi))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))))))))))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_add, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_g, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x))))) => ![V_m, V_g, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (![V_x]: (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))) & p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x))))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel))))))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g)))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) & p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) & (p(s(t_bool,V_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_combins_Iu_u_ou_u_IDu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_combins_Cu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_x, V_x0, V_x1]: s(TV_u_27c,h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27b,V_x0),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,V_x0)))).
fof(ah4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) & p(s(t_bool,V_C))) | p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) | p(s(t_bool,V_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_predu_u_sets_FINITEu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & ![V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_e]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))))))) => ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_f]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_THMu_c1, axiom, ![TV_u_27a]: ![V_s, V_f, V_e]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_e))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_e))))))))).
fof(ah4s_predu_u_sets_DELETEu_u_NONu_u_ELEMENT, axiom, ![TV_u_27a]: ![V_x, V_s]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) <=> s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),V_s))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_combins_Su_u_DEF, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_x0, V_x1]: s(TV_u_27c,h4s_combins_s(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
fof(ah4s_extreals_addu_u_rzero, axiom, ![V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_extreals_extreal,V_x)).
fof(ah4s_predu_u_sets_INu_u_SING, axiom, ![TV_u_27a]: ![V_y, V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f0))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_lebesgues_psfisu_u_add, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_g, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x))))) => ![V_m, V_g, V_f, V_b, V_a]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,V_a),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f)))))) & p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,V_b),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g)))))))) => p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_a),s(t_h4s_extreals_extreal,V_b))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))))))))))).
fof(ah4s_lebesgues_psfisu_u_zero, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_uu_0),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) => ![V_m, V_a]: (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) => (p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,V_a),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_uu_0)))))) <=> s(t_h4s_extreals_extreal,V_a) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_add, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_g, V_x0]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))),s(TV_u_27a,V_x0))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x0))))) => ![V_y, V_x, V_su_27, V_s, V_m, V_g, V_f, V_b, V_a]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x)))) & p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(t_fun(t_h4s_nums_num,t_bool),V_su_27),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_b),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y)))))) => ?[V_su_27u_27, V_c, V_z]: (p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))),s(t_fun(t_h4s_nums_num,t_bool),V_su_27u_27),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_z)))) & s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_su_27),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_b),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y))))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_su_27u_27),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_z))))))).
fof(ah4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_present, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_z, V_c, V_t, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_z))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c))),s(TV_u_27a,V_t))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_z),s(t_h4s_nums_num,V_i))))),s(t_h4s_extreals_extreal,h4s_measures_indicatoru_u_fn(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_t))))) => ![V_y, V_x, V_su_27, V_s, V_m, V_g, V_f, V_b, V_a]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x)))) & p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(t_fun(t_h4s_nums_num,t_bool),V_su_27),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_b),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y)))))) => ?[V_z, V_zu_27, V_c, V_k]: (![V_t]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_t),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_t))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_z))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c))),s(TV_u_27a,V_t))),s(t_fun(t_h4s_nums_num,t_bool),V_k)))) & (![V_t]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_t),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_t))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_zu_27))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c))),s(TV_u_27a,V_t))),s(t_fun(t_h4s_nums_num,t_bool),V_k)))) & (s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_k),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_z))) & (s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_su_27),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_b),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_k),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_zu_27))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_k)))) & (![V_i]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),V_k)))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_z),s(t_h4s_nums_num,V_i))))))) & (![V_i]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),V_k)))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_zu_27),s(t_h4s_nums_num,V_i))))))) & (![V_i, V_j]: ((p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),V_k)))) & (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_j),s(t_fun(t_h4s_nums_num,t_bool),V_k)))) & ~ (s(t_h4s_nums_num,V_i) = s(t_h4s_nums_num,V_j)))) => p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c),s(t_h4s_nums_num,V_i))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c),s(t_h4s_nums_num,V_j))))))) & (![V_i]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),V_k)))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c),s(t_h4s_nums_num,V_i))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))) & s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_c),s(t_fun(t_h4s_nums_num,t_bool),V_k))))) = s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))))))))))).
fof(ah4s_predu_u_sets_EMPTYu_u_DELETE, axiom, ![TV_u_27a]: ![V_x]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_lebesgues_lemmau_u_fnu_u_monou_u_increasing, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_m, V_f, V_x, V_n]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_lebesgues_fnu_u_seq(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) => ![V_x, V_m, V_f]: (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))) => p(s(t_bool,h4s_extreals_extu_u_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x))))))))).
fof(ah4s_lebesgues_lemmau_u_fnu_u_sup, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_m, V_f, V_x, V_n]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_lebesgues_fnu_u_seq(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) => ![V_x, V_m, V_f]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))))))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_lebesgues_lemmau_u_fnu_u_5, axiom, ![TV_u_27a]: ![V_x, V_n, V_m, V_f]: (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_lebesgues_fnu_u_seq(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x)))))))).
fof(ah4s_lebesgues_lemmau_u_fnu_u_inu_u_psfis, axiom, ![TV_u_27a]: ![V_n, V_m, V_f]: ((![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))) & (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))) => p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,h4s_lebesgues_fnu_u_sequ_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_lebesgues_fnu_u_seq(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n)))))))))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f0)).
fof(ah4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_addu_u_alt, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_y, V_i]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y))),s(t_h4s_nums_num,V_i))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_nums_num,V_i))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y),s(t_h4s_nums_num,V_i))))) => ![V_y, V_x, V_s, V_m, V_g, V_f, V_a]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x)))) & p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y)))))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y))))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y)))))))).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_measures_posu_u_simpleu_u_fnu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_a, V_t, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a))),s(TV_u_27a,V_t))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_nums_num,V_i))))),s(t_h4s_extreals_extreal,h4s_measures_indicatoru_u_fn(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_t))))) => ![V_x, V_s, V_m, V_f, V_a]: (p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x)))) <=> (![V_t]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_t)))))) & (![V_t]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_t),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_t))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a))),s(TV_u_27a,V_t))),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) & (![V_i]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_h4s_nums_num,V_i))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) & (![V_i]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_nums_num,V_i))))))) & (![V_i, V_j]: ((p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) & (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_j),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) & ~ (s(t_h4s_nums_num,V_i) = s(t_h4s_nums_num,V_j)))) => p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_h4s_nums_num,V_i))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_h4s_nums_num,V_j))))))) & s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))))))).
fof(ah4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_0, axiom, ![TV_u_27a]: ![V_s, V_f]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_extreals_leu_u_refl, axiom, ![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_x))))).
fof(ah4s_extreals_mulu_u_lzero, axiom, ![V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_f]: s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_lebesgues_posu_u_simpleu_u_fnu_u_addu_u_alt, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_x, V_y, V_i]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_1),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y))),s(t_h4s_nums_num,V_i))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_nums_num,V_i))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y),s(t_h4s_nums_num,V_i))))) => ![V_uu_0]: (![V_f, V_g, V_t]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))),s(TV_u_27a,V_t))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_t))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_t))))) => ![V_y, V_x, V_s, V_m, V_g, V_f, V_a]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x)))) & p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y)))))) => p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_1),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y)))))))))).
fof(ah4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_THMu_c1, axiom, ![TV_u_27a]: ![V_s, V_f, V_e]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(TV_u_27a,V_e))),s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_e))))))))).
fof(ah4s_extreals_extrealu_u_ofu_u_numu_u_def, axiom, ![V_n]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n)))))).
fof(ah4s_reals_leu_u_intu_c0, axiom, ![V_n, V_m]: s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_NOTu_u_LEQ, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_reals_REALu_u_ADDu_u_RID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> s(t_bool,V_t) = s(t_bool,f0))).
fof(ah4s_lebesgues_INu_u_psfisu_u_eq, axiom, ![TV_u_27a]: ![V_r, V_m, V_f]: (p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,V_r),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f)))))) <=> ?[V_s, V_a, V_x]: (p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x)))) & s(t_h4s_extreals_extreal,V_r) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x)))))).
fof(ah4s_measures_INu_u_MEASURABLEu_u_BORELu_u_POSu_u_SIMPLEu_u_FN, axiom, ![TV_u_27a]: ![V_m, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & ?[V_s, V_a, V_x]: p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))).
fof(ah4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_lebesgues_lemmau_u_fnu_u_upperu_u_bounded, axiom, ![TV_u_27a]: ![V_x, V_n, V_m, V_f]: (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_lebesgues_fnu_u_seq(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_measures_INu_u_MEASURABLEu_u_BORELu_u_FNu_u_MINUS, axiom, ![TV_u_27a]: ![V_f, V_a]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_minus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))).
fof(ah4s_measures_INu_u_MEASURABLEu_u_BORELu_u_FNu_u_PLUS, axiom, ![TV_u_27a]: ![V_f, V_a]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))).
fof(ah4s_measures_FNu_u_MINUSu_u_POS, axiom, ![TV_u_27a]: ![V_x, V_g]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_minus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))),s(TV_u_27a,V_x))))))).
fof(ah4s_measures_FNu_u_PLUSu_u_POS, axiom, ![TV_u_27a]: ![V_x, V_g]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))),s(TV_u_27a,V_x))))))).
fof(ah4s_lebesgues_lebesgueu_u_monotoneu_u_convergenceu_u_lemma, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_m, V_fi, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))))) => ![V_uu_0]: (![V_fi, V_x, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))) => ![V_ru_27, V_m, V_g, V_fi, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (![V_i]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))) & (![V_x]: p(s(t_bool,h4s_extreals_extu_u_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x)))))) & (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))) & (p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,V_ru_27),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g)))))) & (![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))) & ![V_i, V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x)))))))))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_ru_27),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ)))))))))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_monou_u_mspace, axiom, ![TV_u_27a]: ![V_m, V_g, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))))))) & (![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))) & ![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x))))))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))),s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f)))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_m, V_f, V_r]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_g]: (p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,V_r),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g)))))) & ![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))))) & s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool),happ(s(t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool)))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_r))) = s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool),h4s_pairs_u_2c(s(t_h4s_extreals_extreal,V_r),s(t_bool,V_v)))) => ![V_m, V_f]: s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool)))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))))))).
fof(ah4s_extreals_supu_u_le, axiom, ![V_x, V_p]: (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),V_p))),s(t_h4s_extreals_extreal,V_x)))) <=> ![V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_extreals_extreal,t_bool),V_p),s(t_h4s_extreals_extreal,V_y)))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_x))))))).
fof(ah4s_extreals_leu_u_antisym, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y)))) & p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_x))))) <=> s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,V_y))).
fof(ah4s_predu_u_sets_INu_u_UNIV, axiom, ![TV_u_27a]: ![V_x]: p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))).
fof(ah4s_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
fof(ah4s_predu_u_sets_INu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> ?[V_x]: (s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_pairs_CLOSEDu_u_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_markers_Abbrevu_u_def, axiom, ![V_x]: s(t_bool,h4s_markers_abbrev(s(t_bool,V_x))) = s(t_bool,V_x)).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_pos, axiom, ![TV_u_27a]: ![V_m, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & ![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f)))))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_posu_u_simpleu_u_fn, axiom, ![TV_u_27a]: ![V_x, V_s, V_m, V_f, V_a]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_mono, axiom, ![TV_u_27a]: ![V_m, V_g, V_f]: (![V_x]: (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))) & p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x))))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g)))))))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_measures_FNu_u_PLUSu_u_POSu_u_ID, axiom, ![TV_u_27a]: ![V_g]: (![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x)))))) => s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))) = s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))).
fof(ah4s_extreals_supu_u_addu_u_mono, axiom, ![V_uu_0]: (![V_f, V_g, V_n]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),t_fun(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_g))),s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_g),s(t_h4s_nums_num,V_n))))) => ![V_g, V_f]: ((![V_n]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n)))))) & (![V_n]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) & (![V_n]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_g),s(t_h4s_nums_num,V_n)))))) & ![V_n]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_g),s(t_h4s_nums_num,V_n))),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_g),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))))))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),t_fun(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_g))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_g),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ)))))))))).
fof(ah4s_extreals_extu_u_monou_u_increasingu_u_def, axiom, ![V_f]: (p(s(t_bool,h4s_extreals_extu_u_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f)))) <=> ![V_m, V_n]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_m))),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))))))))).
fof(ah4s_extreals_leu_u_add2, axiom, ![V_z, V_y, V_x, V_w]: ((p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_w),s(t_h4s_extreals_extreal,V_x)))) & p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_z))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_w),s(t_h4s_extreals_extreal,V_y))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_z)))))))).
fof(ah4s_extreals_leu_u_add, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,V_x)))) & p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,V_y))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y)))))))).
fof(ah4s_extreals_extu_u_monou_u_increasingu_u_suc, axiom, ![V_f]: (p(s(t_bool,h4s_extreals_extu_u_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f)))) <=> ![V_n]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))))).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ch4s_lebesgues_posu_u_fnu_u_integralu_u_sum, conjecture, ![TV_u_27a,TV_u_27b]: ![V_uu_2]: (![V_m, V_f, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27b,t_h4s_extreals_extreal))),V_uu_2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(TV_u_27b,V_i))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(TV_u_27b,V_i))))) => ![V_uu_1]: (![V_f, V_x, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(TV_u_27a,V_x))),s(TV_u_27b,V_i))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(TV_u_27b,V_i))),s(TV_u_27a,V_x))) => ![V_uu_0]: (![V_f, V_s, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(t_fun(TV_u_27b,t_bool),V_s))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_s))) => ![V_s, V_m, V_f]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),V_s)))) & (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (![V_i]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_i),s(t_fun(TV_u_27b,t_bool),V_s)))) => ![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(TV_u_27b,V_i))),s(TV_u_27a,V_x))))))) & ![V_i]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_i),s(t_fun(TV_u_27b,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(TV_u_27b,V_i))),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(t_fun(TV_u_27b,t_bool),V_s))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27b,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27b,t_h4s_extreals_extreal))),V_uu_2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(t_fun(TV_u_27b,t_bool),V_s)))))))).
