%   ORIGINAL: h4/lebesgue/lebesgue__monotone__convergence__lemma
% 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/measure/IN__MEASURABLE__BOREL__MONO__SUP: !fn f a. h4/measure/sigma__algebra a /\ (!n. h4/bool/IN (fn n) (h4/measure/measurable a h4/measure/Borel)) /\ (!n x. h4/bool/IN x (h4/measure/space a) ==> h4/extreal/extreal__le (fn n x) (fn (h4/num/SUC n) x)) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> f x = h4/extreal/extreal__sup (h4/pred__set/IMAGE (\n. fn n x) h4/pred__set/UNIV)) ==> h4/bool/IN f (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/lebesgue/integrable__def: !m f. h4/lebesgue/integrable m f <=> 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 (h4/measure/fn__plus f) = h4/extreal/PosInf) /\ ~(h4/lebesgue/pos__fn__integral m (h4/measure/fn__minus f) = h4/extreal/PosInf)
% Assm: h4/lebesgue/psfis__mono: !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) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__le (f x) (g x)) ==> h4/extreal/extreal__le a b
% 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/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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/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/measure/IN__MEASURABLE__BOREL__ALL__MEASURE: !m f. h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__lt (f x) c))) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__le c (f x)))) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__le (f x) c))) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__lt c (f x)))) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__lt c (f x) /\ h4/extreal/extreal__lt (f x) d))) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__le c (f x) /\ h4/extreal/extreal__lt (f x) d))) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__lt c (f x) /\ h4/extreal/extreal__le (f x) d))) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__le c (f x) /\ h4/extreal/extreal__le (f x) d))) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (f x = c))) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (~(f x = c)))) (h4/measure/m__space m)) (h4/measure/measurable__sets m))
% 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/lebesgue/psfis__present: !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) ==> (?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) /\ a = h4/lebesgue/pos__simple__fn__integral m k c z /\ b = 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/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/lebesgue/psfis__pos: !m f a. h4/measure/measure__space m /\ h4/bool/IN a (h4/lebesgue/psfis m f) ==> (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x))
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% 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/lebesgue/psfis__sub: !m g f b a. h4/measure/measure__space m /\ (!x. h4/extreal/extreal__le (g x) (f x)) /\ (!x. ~(g x = h4/extreal/PosInf)) /\ h4/bool/IN a (h4/lebesgue/psfis m f) /\ h4/bool/IN b (h4/lebesgue/psfis m g) ==> h4/bool/IN (h4/extreal/extreal__sub a b) (h4/lebesgue/psfis m (\x. h4/extreal/extreal__sub (f x) (g 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/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% 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/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/lebesgue/psfis__def: !m f. h4/lebesgue/psfis m f = h4/pred__set/IMAGE (h4/pair/UNCURRY (\s. h4/pair/UNCURRY (\a x. h4/lebesgue/pos__simple__fn__integral m s a x))) (h4/lebesgue/psfs m f)
% Assm: h4/lebesgue/psfs__def: !m f. h4/lebesgue/psfs m f = h4/pred__set/GSPEC (h4/pair/UNCURRY (\s. h4/pair/UNCURRY (\a x. h4/pair/_2C (h4/pair/_2C s (h4/pair/_2C a x)) (h4/measure/pos__simple__fn m f s a x))))
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/measure/subsets__def: !y x. h4/measure/subsets (h4/pair/_2C x y) = y
% Assm: h4/measure/space__def: !y x. h4/measure/space (h4/pair/_2C x y) = x
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% 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/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/C__DEF: h4/combin/C = (\f x y. f y x)
% Assm: h4/combin/S__DEF: h4/combin/S = (\f g x. f x (g x))
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/combin/I__THM: !x. h4/combin/I x = 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/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/util__prob/IN__BIGINTER__IMAGE: !x s f. h4/bool/IN x (h4/pred__set/BIGINTER (h4/pred__set/IMAGE f s)) <=> (!y. h4/bool/IN y s ==> h4/bool/IN x (f y))
% Assm: h4/util__prob/IN__FUNSET: !f Q P. h4/bool/IN f (h4/util__prob/FUNSET P Q) <=> (!x. h4/bool/IN x P ==> h4/bool/IN (f x) Q)
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pred__set/IN__INTER: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/measure/SPACE: !a. h4/pair/_2C (h4/measure/space a) (h4/measure/subsets a) = a
% Assm: h4/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% Assm: h4/measure/SIGMA__ALGEBRA__FN__BIGINTER: !a. h4/measure/sigma__algebra a ==> h4/measure/subset__class (h4/measure/space a) (h4/measure/subsets a) /\ h4/bool/IN h4/pred__set/EMPTY (h4/measure/subsets a) /\ (!s. h4/bool/IN s (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/DIFF (h4/measure/space a) s) (h4/measure/subsets a)) /\ (!f. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (h4/measure/subsets a)) ==> h4/bool/IN (h4/pred__set/BIGINTER (h4/pred__set/IMAGE f h4/pred__set/UNIV)) (h4/measure/subsets a))
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/measure/IN__MEASURABLE__BOREL__ALT2: !f a. h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) <=> h4/measure/sigma__algebra a /\ h4/bool/IN f (h4/util__prob/FUNSET (h4/measure/space a) h4/pred__set/UNIV) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__le (f x) c))) (h4/measure/space a)) (h4/measure/subsets a))
% 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/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% 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/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/lebesgue/pos__simple__fn__integral__sub: !y x s_27 s m g f b a. h4/measure/measure__space m /\ (!x0. h4/extreal/extreal__le (g x0) (f x0)) /\ (!x0. ~(g x0 = h4/extreal/PosInf)) /\ 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__sub (f x0) (g x0)) s_27_27 c z /\ h4/extreal/extreal__sub (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/bool/CONJ__COMM: !t2 t1. t1 /\ t2 <=> t2 /\ t1
% Assm: h4/lebesgue/pos__simple__fn__integral__mono: !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 /\ (!x0. h4/bool/IN x0 (h4/measure/m__space m) ==> h4/extreal/extreal__le (f x0) (g x0)) ==> h4/extreal/extreal__le (h4/lebesgue/pos__simple__fn__integral m s a x) (h4/lebesgue/pos__simple__fn__integral m s_27 b y)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/measure/IN__MEASURABLE__BOREL__ALL: !f a. h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) ==> (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__lt (f x) c))) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__le c (f x)))) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__le (f x) c))) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__lt c (f x)))) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__lt c (f x) /\ h4/extreal/extreal__lt (f x) d))) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__le c (f x) /\ h4/extreal/extreal__lt (f x) d))) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__lt c (f x) /\ h4/extreal/extreal__le (f x) d))) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__le c (f x) /\ h4/extreal/extreal__le (f x) d))) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (~(f x = c)))) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (\x. h4/pair/_2C x (f x = c))) (h4/measure/space a)) (h4/measure/subsets a))
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/measure/IN__MEASURABLE__BOREL__CMUL: !z g f a. h4/measure/sigma__algebra a /\ h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> g x = h4/extreal/extreal__mul (h4/extreal/Normal z) (f x)) ==> h4/bool/IN g (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/measure/IN__MEASURABLE__BOREL__INDICATOR: !f a A. h4/measure/sigma__algebra a /\ h4/bool/IN A (h4/measure/subsets a) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> f x = h4/measure/indicator__fn A x) ==> h4/bool/IN f (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/measure/IN__MEASURABLE__BOREL__SUM: !s g f a. h4/pred__set/FINITE s /\ h4/measure/sigma__algebra a /\ (!i. h4/bool/IN i s ==> h4/bool/IN (f i) (h4/measure/measurable a h4/measure/Borel)) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> g x = h4/extreal/EXTREAL__SUM__IMAGE (\i. f i x) s) ==> h4/bool/IN g (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/measure/measure__space__def: !m. h4/measure/measure__space m <=> h4/measure/sigma__algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/countably__additive m
% Assm: h4/extreal/le__refl: !x. h4/extreal/extreal__le x x
% Assm: h4/numeral/numeral__lte_c2: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__add_c0: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = 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/numeral/numeral__distrib_c2: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% 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/LESS__EQ__EXISTS: !n m. h4/arithmetic/_3C_3D m n <=> (?p. n = h4/arithmetic/_2B m p)
% 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/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/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/arithmetic/LESS__EQ__ADD: !n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/arithmetic/ADD__CLAUSES_c3: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/extreal/le__trans: !z y x. h4/extreal/extreal__le x y /\ h4/extreal/extreal__le y z ==> h4/extreal/extreal__le x z
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/extreal/extreal__case__def_c2: !v1 v f a. h4/extreal/extreal__CASE (h4/extreal/Normal a) v v1 f = f a
% Assm: h4/extreal/extreal__le__def_c1: !a. h4/extreal/extreal__le h4/extreal/NegInf a <=> T
% Assm: h4/extreal/extreal__le__def_c0: !y x. h4/extreal/extreal__le (h4/extreal/Normal x) (h4/extreal/Normal y) <=> h4/real/real__lte x y
% Assm: h4/extreal/extreal__case__def_c1: !v1 v f. h4/extreal/extreal__CASE h4/extreal/PosInf v v1 f = v1
% 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/extreal/Normal0: h4/extreal/Normal = h4/extreal/_20_40ind__typeextreal2
% Assm: h4/extreal/PosInf0: h4/extreal/PosInf = h4/extreal/_20_40ind__typeextreal1
% Assm: h4/extreal/extreal__le__def_c2: h4/extreal/extreal__le h4/extreal/PosInf h4/extreal/PosInf <=> T
% Assm: h4/extreal/extreal__case__def_c0: !v1 v f. h4/extreal/extreal__CASE h4/extreal/NegInf v v1 f = v
% Assm: h4/extreal/NegInf0: h4/extreal/NegInf = h4/extreal/_20_40ind__typeextreal0
% Goal: !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))
%   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_measures_INu_u_MEASURABLEu_u_BORELu_u_MONOu_u_SUP]: !_0. (!fn x n. happ (happ (happ _0 fn) x) n = happ (happ fn n) x) ==> (!fn f a. h4/measure/sigma__algebra a /\ (!n. h4/bool/IN (happ fn n) (h4/measure/measurable a h4/measure/Borel)) /\ (!n x. h4/bool/IN x (h4/measure/space a) ==> h4/extreal/extreal__le (happ (happ fn n) x) (happ (happ fn (h4/num/SUC n)) x)) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> happ f x = h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 fn) x) h4/pred__set/UNIV)) ==> h4/bool/IN f (h4/measure/measurable a h4/measure/Borel))
% Assm [h4s_lebesgues_integrableu_u_def]: !m f. h4/lebesgue/integrable m f <=> 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 (h4/measure/fn__plus f) = h4/extreal/PosInf) /\ ~(h4/lebesgue/pos__fn__integral m (h4/measure/fn__minus f) = h4/extreal/PosInf)
% Assm [h4s_lebesgues_psfisu_u_mono]: !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) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__le (happ f x) (happ g x)) ==> h4/extreal/extreal__le a b
% 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_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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_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_measures_INu_u_MEASURABLEu_u_BORELu_u_ALLu_u_MEASURE]: !_9. (!f c x. ?v. (v <=> ~(happ f x = c)) /\ happ (happ (happ _9 f) c) x = h4/pair/_2C x v) ==> (!_8. (!f c x. ?v. (v <=> happ f x = c) /\ happ (happ (happ _8 f) c) x = h4/pair/_2C x v) ==> (!_7. (!c f d x. ?v. (v <=> h4/extreal/extreal__le c (happ f x) /\ h4/extreal/extreal__le (happ f x) d) /\ happ (happ (happ (happ _7 c) f) d) x = h4/pair/_2C x v) ==> (!_6. (!c f d x. ?v. (v <=> h4/extreal/extreal__lt c (happ f x) /\ h4/extreal/extreal__le (happ f x) d) /\ happ (happ (happ (happ _6 c) f) d) x = h4/pair/_2C x v) ==> (!_5. (!c f d x. ?v. (v <=> h4/extreal/extreal__le c (happ f x) /\ h4/extreal/extreal__lt (happ f x) d) /\ happ (happ (happ (happ _5 c) f) d) x = h4/pair/_2C x v) ==> (!_4. (!c f d x. ?v. (v <=> h4/extreal/extreal__lt c (happ f x) /\ h4/extreal/extreal__lt (happ f x) d) /\ happ (happ (happ (happ _4 c) f) d) x = h4/pair/_2C x v) ==> (!_3. (!c f x. happ (happ (happ _3 c) f) x = h4/pair/_2C x (h4/extreal/extreal__lt c (happ f x))) ==> (!_2. (!f c x. happ (happ (happ _2 f) c) x = h4/pair/_2C x (h4/extreal/extreal__le (happ f x) c)) ==> (!_1. (!c f x. happ (happ (happ _1 c) f) x = h4/pair/_2C x (h4/extreal/extreal__le c (happ f x))) ==> (!_0. (!f c x. happ (happ (happ _0 f) c) x = h4/pair/_2C x (h4/extreal/extreal__lt (happ f x) c)) ==> (!m f. h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _0 f) c)) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _1 c) f)) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _2 f) c)) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _3 c) f)) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ (happ _4 c) f) d)) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ (happ _5 c) f) d)) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ (happ _6 c) f) d)) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ (happ _7 c) f) d)) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _8 f) c)) (h4/measure/m__space m)) (h4/measure/measurable__sets m)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _9 f) c)) (h4/measure/m__space m)) (h4/measure/measurable__sets m))))))))))))
% 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_lebesgues_psfisu_u_present]: !_0. (!z c t i. happ (happ (happ (happ _0 z) c) t) i = h4/extreal/extreal__mul (happ h4/extreal/Normal (happ z i)) (h4/measure/indicator__fn (happ c i) t)) ==> (!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) ==> (?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) /\ a = h4/lebesgue/pos__simple__fn__integral m k c z /\ b = 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_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_lebesgues_psfisu_u_pos]: !m f a. h4/measure/measure__space m /\ h4/bool/IN a (h4/lebesgue/psfis m f) ==> (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x))
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% 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_lebesgues_psfisu_u_sub]: !_0. (!f g x. happ (happ (happ _0 f) g) x = h4/extreal/extreal__sub (happ f x) (happ g x)) ==> (!m g f b a. h4/measure/measure__space m /\ (!x. h4/extreal/extreal__le (happ g x) (happ f x)) /\ (!x. ~(happ g x = h4/extreal/PosInf)) /\ h4/bool/IN a (h4/lebesgue/psfis m f) /\ h4/bool/IN b (h4/lebesgue/psfis m g) ==> h4/bool/IN (h4/extreal/extreal__sub a b) (h4/lebesgue/psfis m (happ (happ _0 f) g)))
% 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_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% 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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% 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_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_lebesgues_psfisu_u_def]: !_2. (!m s a x. happ (happ (happ (happ _2 m) s) a) x = h4/lebesgue/pos__simple__fn__integral m s a x) ==> (!_1. (!m s a. happ (happ (happ _1 m) s) a = happ (happ (happ _2 m) s) a) ==> (!_0. (!m s. happ (happ _0 m) s = h4/pair/UNCURRY (happ (happ _1 m) s)) ==> (!m f. h4/lebesgue/psfis m f = h4/pred__set/IMAGE (h4/pair/UNCURRY (happ _0 m)) (h4/lebesgue/psfs m f))))
% Assm [h4s_lebesgues_psfsu_u_def]: !_2. (!m f s a x. happ (happ (happ (happ (happ _2 m) f) s) a) x = h4/pair/_2C (h4/pair/_2C s (h4/pair/_2C a x)) (h4/measure/pos__simple__fn m f s a x)) ==> (!_1. (!m f s a. happ (happ (happ (happ _1 m) f) s) a = happ (happ (happ (happ _2 m) f) s) a) ==> (!_0. (!m f s. happ (happ (happ _0 m) f) s = h4/pair/UNCURRY (happ (happ (happ _1 m) f) s)) ==> (!m f. h4/lebesgue/psfs m f = h4/pred__set/GSPEC (h4/pair/UNCURRY (happ (happ _0 m) f)))))
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_measures_subsetsu_u_def]: !y x. h4/measure/subsets (h4/pair/_2C x y) = y
% Assm [h4s_measures_spaceu_u_def]: !y x. h4/measure/space (h4/pair/_2C x y) = x
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% 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_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_Cu_u_DEF]: !x x x. h4/combin/C x x x = happ (happ x x) x
% 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_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% 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_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_utilu_u_probs_INu_u_BIGINTERu_u_IMAGE]: !x s f. h4/bool/IN x (h4/pred__set/BIGINTER (h4/pred__set/IMAGE f s)) <=> (!y. h4/bool/IN y s ==> h4/bool/IN x (happ f y))
% Assm [h4s_utilu_u_probs_INu_u_FUNSET]: !f Q P. h4/bool/IN f (h4/util__prob/FUNSET P Q) <=> (!x. h4/bool/IN x P ==> h4/bool/IN (happ f x) Q)
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_INu_u_INTER]: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_measures_SPACE]: !a. h4/pair/_2C (h4/measure/space a) (h4/measure/subsets a) = a
% Assm [h4s_combins_Iu_u_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% Assm [h4s_measures_SIGMAu_u_ALGEBRAu_u_FNu_u_BIGINTER]: !a. h4/measure/sigma__algebra a ==> h4/measure/subset__class (h4/measure/space a) (h4/measure/subsets a) /\ h4/bool/IN h4/pred__set/EMPTY (h4/measure/subsets a) /\ (!s. h4/bool/IN s (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/DIFF (h4/measure/space a) s) (h4/measure/subsets a)) /\ (!f. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (h4/measure/subsets a)) ==> h4/bool/IN (h4/pred__set/BIGINTER (h4/pred__set/IMAGE f h4/pred__set/UNIV)) (h4/measure/subsets a))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_ALT2]: !_0. (!f c x. happ (happ (happ _0 f) c) x = h4/pair/_2C x (h4/extreal/extreal__le (happ f x) c)) ==> (!f a. h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) <=> h4/measure/sigma__algebra a /\ h4/bool/IN f (h4/util__prob/FUNSET (h4/measure/space a) h4/pred__set/UNIV) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _0 f) c)) (h4/measure/space a)) (h4/measure/subsets a)))
% 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 (happ 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_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% 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 (happ 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_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_sub]: !_0. (!f g x0. happ (happ (happ _0 f) g) x0 = h4/extreal/extreal__sub (happ f x0) (happ g x0)) ==> (!y x s_27 s m g f b a. h4/measure/measure__space m /\ (!x0. h4/extreal/extreal__le (happ g x0) (happ f x0)) /\ (!x0. ~(happ g x0 = h4/extreal/PosInf)) /\ 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__sub (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_bools_CONJu_u_COMM]: !t2 t1. t1 /\ t2 <=> t2 /\ t1
% Assm [h4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_mono]: !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 /\ (!x0. h4/bool/IN x0 (h4/measure/m__space m) ==> h4/extreal/extreal__le (happ f x0) (happ g x0)) ==> h4/extreal/extreal__le (h4/lebesgue/pos__simple__fn__integral m s a x) (h4/lebesgue/pos__simple__fn__integral m s_27 b y)
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% 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_measures_INu_u_MEASURABLEu_u_BORELu_u_ALL]: !_9. (!f c x. ?v. (v <=> happ f x = c) /\ happ (happ (happ _9 f) c) x = h4/pair/_2C x v) ==> (!_8. (!f c x. ?v. (v <=> ~(happ f x = c)) /\ happ (happ (happ _8 f) c) x = h4/pair/_2C x v) ==> (!_7. (!c f d x. ?v. (v <=> h4/extreal/extreal__le c (happ f x) /\ h4/extreal/extreal__le (happ f x) d) /\ happ (happ (happ (happ _7 c) f) d) x = h4/pair/_2C x v) ==> (!_6. (!c f d x. ?v. (v <=> h4/extreal/extreal__lt c (happ f x) /\ h4/extreal/extreal__le (happ f x) d) /\ happ (happ (happ (happ _6 c) f) d) x = h4/pair/_2C x v) ==> (!_5. (!c f d x. ?v. (v <=> h4/extreal/extreal__le c (happ f x) /\ h4/extreal/extreal__lt (happ f x) d) /\ happ (happ (happ (happ _5 c) f) d) x = h4/pair/_2C x v) ==> (!_4. (!c f d x. ?v. (v <=> h4/extreal/extreal__lt c (happ f x) /\ h4/extreal/extreal__lt (happ f x) d) /\ happ (happ (happ (happ _4 c) f) d) x = h4/pair/_2C x v) ==> (!_3. (!c f x. happ (happ (happ _3 c) f) x = h4/pair/_2C x (h4/extreal/extreal__lt c (happ f x))) ==> (!_2. (!f c x. happ (happ (happ _2 f) c) x = h4/pair/_2C x (h4/extreal/extreal__le (happ f x) c)) ==> (!_1. (!c f x. happ (happ (happ _1 c) f) x = h4/pair/_2C x (h4/extreal/extreal__le c (happ f x))) ==> (!_0. (!f c x. happ (happ (happ _0 f) c) x = h4/pair/_2C x (h4/extreal/extreal__lt (happ f x) c)) ==> (!f a. h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) ==> (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _0 f) c)) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _1 c) f)) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _2 f) c)) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _3 c) f)) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ (happ _4 c) f) d)) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ (happ _5 c) f) d)) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ (happ _6 c) f) d)) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c d. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ (happ _7 c) f) d)) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _8 f) c)) (h4/measure/space a)) (h4/measure/subsets a)) /\ (!c. h4/bool/IN (h4/pred__set/INTER (h4/pred__set/GSPEC (happ (happ _9 f) c)) (h4/measure/space a)) (h4/measure/subsets a))))))))))))
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_CMUL]: !z g f a. h4/measure/sigma__algebra a /\ h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> happ g x = h4/extreal/extreal__mul (happ h4/extreal/Normal z) (happ f x)) ==> h4/bool/IN g (h4/measure/measurable a h4/measure/Borel)
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_INDICATOR]: !f a A. h4/measure/sigma__algebra a /\ h4/bool/IN A (h4/measure/subsets a) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> happ f x = h4/measure/indicator__fn A x) ==> h4/bool/IN f (h4/measure/measurable a h4/measure/Borel)
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_SUM]: !_0. (!f x i. happ (happ (happ _0 f) x) i = happ (happ f i) x) ==> (!s g f a. h4/pred__set/FINITE s /\ h4/measure/sigma__algebra a /\ (!i. h4/bool/IN i s ==> h4/bool/IN (happ f i) (h4/measure/measurable a h4/measure/Borel)) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> happ g x = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ _0 f) x) s) ==> h4/bool/IN g (h4/measure/measurable a h4/measure/Borel))
% Assm [h4s_measures_measureu_u_spaceu_u_def]: !m. h4/measure/measure__space m <=> h4/measure/sigma__algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/countably__additive m
% Assm [h4s_extreals_leu_u_refl]: !x. h4/extreal/extreal__le x x
% Assm [h4s_numerals_numeralu_u_lteu_c2]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_addu_c0]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = 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_numerals_numeralu_u_distribu_c2]: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% 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_LESSu_u_EQu_u_EXISTS]: !n m. h4/arithmetic/_3C_3D m n <=> (?p. n = h4/arithmetic/_2B m p)
% 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_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_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_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_arithmetics_LESSu_u_EQu_u_ADD]: !n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c3]: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_extreals_leu_u_trans]: !z y x. h4/extreal/extreal__le x y /\ h4/extreal/extreal__le y z ==> h4/extreal/extreal__le x z
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_extreals_extrealu_u_caseu_u_defu_c2]: !v1 v f a. h4/extreal/extreal__CASE (happ h4/extreal/Normal a) v v1 f = happ f a
% Assm [h4s_extreals_extrealu_u_leu_u_defu_c1]: !a. h4/extreal/extreal__le h4/extreal/NegInf a <=> T
% Assm [h4s_extreals_extrealu_u_leu_u_defu_c0]: !y x. h4/extreal/extreal__le (happ h4/extreal/Normal x) (happ h4/extreal/Normal y) <=> h4/real/real__lte x y
% Assm [h4s_extreals_extrealu_u_caseu_u_defu_c1]: !v1 v f. h4/extreal/extreal__CASE h4/extreal/PosInf v v1 f = v1
% Assm [h4s_extreals_extrealu_u_ofu_u_numu_u_def]: !n. h4/extreal/extreal__of__num n = happ h4/extreal/Normal (h4/real/real__of__num n)
% Assm [h4s_extreals_Normal0]: h4/extreal/Normal = h4/extreal/_20_40ind__typeextreal2
% Assm [h4s_extreals_PosInf0]: h4/extreal/PosInf = h4/extreal/_20_40ind__typeextreal1
% Assm [h4s_extreals_extrealu_u_leu_u_defu_c2]: h4/extreal/extreal__le h4/extreal/PosInf h4/extreal/PosInf <=> T
% Assm [h4s_extreals_extrealu_u_caseu_u_defu_c0]: !v1 v f. h4/extreal/extreal__CASE h4/extreal/NegInf v v1 f = v
% Assm [h4s_extreals_NegInf0]: h4/extreal/NegInf = h4/extreal/_20_40ind__typeextreal0
% Goal: !_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))))
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_Q1361348,TV_Q1361344]: ![V_f, V_g]: (![V_x]: s(TV_Q1361344,happ(s(t_fun(TV_Q1361348,TV_Q1361344),V_f),s(TV_Q1361348,V_x))) = s(TV_Q1361344,happ(s(t_fun(TV_Q1361348,TV_Q1361344),V_g),s(TV_Q1361348,V_x))) => s(t_fun(TV_Q1361348,TV_Q1361344),V_f) = s(t_fun(TV_Q1361348,TV_Q1361344),V_g))).
fof(ah4s_measures_INu_u_MEASURABLEu_u_BORELu_u_MONOu_u_SUP, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_fn, 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(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_fn))),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),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fn),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) => ![V_fn, V_f, V_a]: ((p(s(t_bool,h4s_measures_sigmau_u_algebra(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)))) & (![V_n]: 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_fn),s(t_h4s_nums_num,V_n))),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)))))) & (![V_n, 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_space(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)))))) => 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_fn),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),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fn),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))),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_space(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_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(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_fn))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))))))) => 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))))))))).
fof(ah4s_lebesgues_integrableu_u_def, axiom, ![TV_u_27a]: ![V_m, V_f]: (p(s(t_bool,h4s_lebesgues_integrable(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_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),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)) & ~ (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_posinf)))))).
fof(ah4s_lebesgues_psfisu_u_mono, axiom, ![TV_u_27a]: ![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)))))) & ![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_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,V_a),s(t_h4s_extreals_extreal,V_b)))))).
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_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_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_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_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_measures_INu_u_MEASURABLEu_u_BORELu_u_ALLu_u_MEASURE, axiom, ![TV_u_27a]: ![V_uu_9]: (![V_f, V_c, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ~ (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,V_c))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_9),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_uu_8]: (![V_f, V_c, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> 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,V_c)) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_8),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_uu_7]: (![V_c, V_f, V_d, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_c),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,V_d)))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_7),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_uu_6]: (![V_c, V_f, V_d, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_c),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,V_d)))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_6),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_uu_5]: (![V_c, V_f, V_d, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_c),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_lt(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,V_d)))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_5),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_uu_4]: (![V_c, V_f, V_d, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_c),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_lt(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,V_d)))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_4),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_uu_3]: (![V_c, V_f, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_3),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_c),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_uu_2]: (![V_f, V_c, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),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,V_c))))) => ![V_uu_1]: (![V_c, V_f, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_1),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_c),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_uu_0]: (![V_f, V_c, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_extreals_extrealu_u_lt(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,V_c))))) => ![V_m, V_f]: (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_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))))),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)))))) & (![V_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_1),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))),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)))))) & (![V_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))))),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)))))) & (![V_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_3),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))),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)))))) & (![V_c, V_d]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_4),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))))),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)))))) & (![V_c, V_d]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_5),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))))),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)))))) & (![V_c, V_d]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_6),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))))),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)))))) & (![V_c, V_d]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_7),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))))),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)))))) & (![V_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_8),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))))),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)))))) & ![V_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_9),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))))),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))))))))))))))))))))))))))).
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_lebesgues_psfisu_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,happ(s(t_fun(t_h4s_realaxs_real,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_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)))))))) => ?[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,V_a) = 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,V_b) = 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_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_lebesgues_psfisu_u_pos, axiom, ![TV_u_27a]: ![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_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))))))) => ![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,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))))))))).
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_bools_TRUTH, axiom, p(s(t_bool,t))).
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_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_lebesgues_psfisu_u_sub, 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_sub(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)))) & (![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_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))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)) & (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_sub(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_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_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),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(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_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
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_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => 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_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_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_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_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_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_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_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_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_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => 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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_lebesgues_psfisu_u_def, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_m, V_s, V_a, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),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(t_h4s_nums_num,t_h4s_realaxs_real),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(t_h4s_nums_num,t_h4s_realaxs_real),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(t_h4s_nums_num,t_h4s_realaxs_real),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(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_x))) => ![V_uu_1]: (![V_m, V_s, V_a]: s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),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(t_h4s_nums_num,t_h4s_realaxs_real),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(t_h4s_nums_num,t_h4s_realaxs_real),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(t_h4s_nums_num,t_h4s_realaxs_real),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_bool),V_s))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a))) = s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),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(t_h4s_nums_num,t_h4s_realaxs_real),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(t_h4s_nums_num,t_h4s_realaxs_real),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(t_h4s_nums_num,t_h4s_realaxs_real),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(t_h4s_nums_num,t_bool),V_s))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a))) => ![V_uu_0]: (![V_m, V_s]: s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),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_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),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(t_h4s_nums_num,t_bool),V_s))) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_extreals_extreal),h4s_pairs_uncurry(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),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(t_h4s_nums_num,t_h4s_realaxs_real),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(t_h4s_nums_num,t_h4s_realaxs_real),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_bool),V_s))))) => ![V_m, V_f]: 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))) = s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_h4s_extreals_extreal),h4s_pairs_uncurry(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),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_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),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(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool),h4s_lebesgues_psfs(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_psfsu_u_def, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_m, V_f, V_s, V_a, V_x]: s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool))),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(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),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(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),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_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(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)))))),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_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_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),h4s_pairs_u_2c(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_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_uu_1]: (![V_m, V_f, V_s, V_a]: s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool))),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(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),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(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),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_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(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)))))),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(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_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool))),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(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),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(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),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_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(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)))))),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_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))) => ![V_uu_0]: (![V_m, V_f, V_s]: s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),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_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),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_fun(t_h4s_nums_num,t_bool),V_s))) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)),h4s_pairs_uncurry(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool))),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(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),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(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),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_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(t_h4s_nums_num,t_h4s_realaxs_real),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)))))),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(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) => ![V_m, V_f]: s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool),h4s_lebesgues_psfs(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_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool)),h4s_pairs_uncurry(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),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_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),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_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_pairs_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_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
fof(ah4s_measures_subsetsu_u_def, axiom, ![TV_u_27a]: ![V_y, V_x]: s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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),V_x),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y)).
fof(ah4s_measures_spaceu_u_def, axiom, ![TV_u_27a]: ![V_y, V_x]: s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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),V_x),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y))))) = s(t_fun(TV_u_27a,t_bool),V_x)).
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_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_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_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_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_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_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_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
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_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_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_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_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_utilu_u_probs_INu_u_BIGINTERu_u_IMAGE, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_s, V_f]: (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_biginter(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(TV_u_27b,t_bool),V_s)))))))) <=> ![V_y]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_y))))))))).
fof(ah4s_utilu_u_probs_INu_u_FUNSET, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_Q, V_P]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),h4s_utilu_u_probs_funset(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_Q))))))).
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_INu_u_INTER, axiom, ![TV_u_27a]: ![V_x, V_t, 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_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_measures_SPACE, axiom, ![TV_u_27a]: ![V_a]: 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_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)).
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_measures_SIGMAu_u_ALGEBRAu_u_FNu_u_BIGINTER, axiom, ![TV_u_27a]: ![V_a]: (p(s(t_bool,h4s_measures_sigmau_u_algebra(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)))) => (p(s(t_bool,h4s_measures_subsetu_u_class(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) & (![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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))))))) & ![V_f]: (p(s(t_bool,h4s_bools_in(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_bool),h4s_utilu_u_probs_funset(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_biginter(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_f),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))))))))).
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_measures_INu_u_MEASURABLEu_u_BORELu_u_ALT2, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_c, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),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,V_c))))) => ![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_measures_sigmau_u_algebra(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)))) & (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_utilu_u_probs_funset(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_univ)))))) & ![V_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))))),s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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))))))))))).
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,happ(s(t_fun(t_h4s_realaxs_real,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_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_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_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,happ(s(t_fun(t_h4s_realaxs_real,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_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
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_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_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_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_sub, 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_sub(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)))) & (![V_x0]: 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_x0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x0)))))) & (![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))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)) & (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_sub(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_bools_CONJu_u_COMM, 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))))).
fof(ah4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_mono, axiom, ![TV_u_27a]: ![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_x0]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x0),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_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)))))))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(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)))))))).
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_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_measures_INu_u_MEASURABLEu_u_BORELu_u_ALL, axiom, ![TV_u_27a]: ![V_uu_9]: (![V_f, V_c, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> 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,V_c)) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_9),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_uu_8]: (![V_f, V_c, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ~ (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,V_c))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_8),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_uu_7]: (![V_c, V_f, V_d, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_c),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,V_d)))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_7),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_uu_6]: (![V_c, V_f, V_d, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_c),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,V_d)))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_6),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_uu_5]: (![V_c, V_f, V_d, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_c),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_lt(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,V_d)))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_5),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_uu_4]: (![V_c, V_f, V_d, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_c),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_lt(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,V_d)))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_4),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_uu_3]: (![V_c, V_f, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_3),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_c),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_uu_2]: (![V_f, V_c, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),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,V_c))))) => ![V_uu_1]: (![V_c, V_f, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_1),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_c),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_uu_0]: (![V_f, V_c, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,h4s_extreals_extrealu_u_lt(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,V_c))))) => ![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)))))) => (![V_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))))),s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) & (![V_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_1),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))),s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) & (![V_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))))),s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) & (![V_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_3),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))),s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) & (![V_c, V_d]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_4),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))))),s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) & (![V_c, V_d]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_5),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))))),s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) & (![V_c, V_d]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_6),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))))),s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) & (![V_c, V_d]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_7),s(t_h4s_extreals_extreal,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_d))))),s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) & (![V_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_8),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))))),s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) & ![V_c]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_9),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_extreals_extreal,V_c))))),s(t_fun(TV_u_27a,t_bool),h4s_measures_space(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_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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))))))))))))))))))))))))))).
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(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_measures_INu_u_MEASURABLEu_u_BORELu_u_CMUL, axiom, ![TV_u_27a]: ![V_z, V_g, V_f, V_a]: ((p(s(t_bool,h4s_measures_sigmau_u_algebra(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)))) & (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)))))) & ![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_space(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_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,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,V_z))),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_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)),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_INDICATOR, axiom, ![TV_u_27a]: ![V_f, V_a, V_A]: ((p(s(t_bool,h4s_measures_sigmau_u_algebra(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)))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_A),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(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)))))) & ![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_space(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_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_measures_indicatoru_u_fn(s(t_fun(TV_u_27a,t_bool),V_A),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)),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_SUM, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![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_0),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_s, V_g, V_f, V_a]: ((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_sigmau_u_algebra(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)))) & (![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)),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))))))) & ![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_space(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_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,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_0),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))))))) => 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)),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_measureu_u_spaceu_u_def, axiom, ![TV_u_27a]: ![V_m]: (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_sigmau_u_algebra(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)))))))) & (p(s(t_bool,h4s_measures_positive(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_countablyu_u_additive(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_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_numerals_numeralu_u_lteu_c2, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f0)).
fof(ah4s_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_addu_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = 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_numerals_numeralu_u_distribu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
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_LESSu_u_EQu_u_EXISTS, 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)))) <=> ?[V_p]: 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))))).
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_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_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_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(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,V_p)))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![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))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_ADD, 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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: 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_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_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(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_bools_RIGHTu_u_EXISTSu_u_ANDu_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_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
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_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c3, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_extreals_leu_u_trans, axiom, ![V_z, 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_z))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_z)))))).
fof(ah4s_bools_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
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_extreals_extrealu_u_caseu_u_defu_c2, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f, V_a]: s(TV_u_27a,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,V_a))),s(TV_u_27a,V_v),s(TV_u_27a,V_v1),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f))) = s(TV_u_27a,happ(s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f),s(t_h4s_realaxs_real,V_a)))).
fof(ah4s_extreals_extrealu_u_leu_u_defu_c1, axiom, ![V_a]: s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,V_a))) = s(t_bool,t)).
fof(ah4s_extreals_extrealu_u_leu_u_defu_c0, axiom, ![V_y, V_x]: s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,V_y))))) = s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))).
fof(ah4s_extreals_extrealu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f]: s(TV_u_27a,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(TV_u_27a,V_v),s(TV_u_27a,V_v1),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f))) = s(TV_u_27a,V_v1)).
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,happ(s(t_fun(t_h4s_realaxs_real,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_extreals_Normal0, axiom, s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal) = s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_u_20u_40indu_u_typeextreal2)).
fof(ah4s_extreals_PosInf0, axiom, s(t_h4s_extreals_extreal,h4s_extreals_posinf) = s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal1)).
fof(ah4s_extreals_extrealu_u_leu_u_defu_c2, axiom, s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,h4s_extreals_posinf))) = s(t_bool,t)).
fof(ah4s_extreals_extrealu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f]: s(TV_u_27a,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(TV_u_27a,V_v),s(TV_u_27a,V_v1),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f))) = s(TV_u_27a,V_v)).
fof(ah4s_extreals_NegInf0, axiom, s(t_h4s_extreals_extreal,h4s_extreals_neginf) = s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal0)).
fof(ch4s_lebesgues_lebesgueu_u_monotoneu_u_convergenceu_u_lemma, conjecture, ![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)))))))))))).
