%   ORIGINAL: h4/probability/joint__conditional
% 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/probability/joint__distribution__le: !p b a Y X. h4/probability/prob__space p /\ h4/probability/events p = h4/pred__set/POW (h4/probability/p__space p) ==> h4/real/real__lte (h4/probability/joint__distribution p X Y (h4/pred__set/CROSS a b)) (h4/probability/distribution p X a)
% Assm: h4/probability/conditional__distribution__def: !p b a Y X. h4/probability/conditional__distribution p X Y a b = h4/real/_2F (h4/probability/joint__distribution p X Y (h4/pred__set/CROSS a b)) (h4/probability/distribution p Y b)
% Assm: h4/probability/joint__distribution__le2: !p b a Y X. h4/probability/prob__space p /\ h4/probability/events p = h4/pred__set/POW (h4/probability/p__space p) ==> h4/real/real__lte (h4/probability/joint__distribution p X Y (h4/pred__set/CROSS a b)) (h4/probability/distribution p Y b)
% Assm: h4/probability/distribution__def: !p X. h4/probability/distribution p X = (\s. h4/probability/prob p (h4/pred__set/INTER (h4/util__prob/PREIMAGE X s) (h4/probability/p__space p)))
% Assm: h4/probability/finite__marginal__product__space__POW2: !s2 s1 p Y X. h4/pred__set/POW (h4/probability/p__space p) = h4/probability/events p /\ h4/probability/random__variable X p (h4/pair/_2C s1 (h4/pred__set/POW s1)) /\ h4/probability/random__variable Y p (h4/pair/_2C s2 (h4/pred__set/POW s2)) /\ h4/pred__set/FINITE (h4/probability/p__space p) /\ h4/pred__set/FINITE s1 /\ h4/pred__set/FINITE s2 ==> h4/measure/measure__space (h4/pair/_2C (h4/pred__set/CROSS s1 s2) (h4/pair/_2C (h4/pred__set/POW (h4/pred__set/CROSS s1 s2)) (h4/probability/joint__distribution p X Y)))
% Assm: h4/probability/joint__distribution__def: !p Y X. h4/probability/joint__distribution p X Y = (\a. h4/probability/prob p (h4/pred__set/INTER (h4/util__prob/PREIMAGE (\x. h4/pair/_2C (X x) (Y x)) a) (h4/probability/p__space p)))
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/TRUTH: T
% 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/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/pred__set/IN__POW: !set e. h4/bool/IN e (h4/pred__set/POW set) <=> h4/pred__set/SUBSET e set
% Assm: h4/pred__set/INTER__SUBSET_c1: !t s. h4/pred__set/SUBSET (h4/pred__set/INTER t s) s
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/util__prob/IN__PREIMAGE: !x s f. h4/bool/IN x (h4/util__prob/PREIMAGE f s) <=> h4/bool/IN (f x) s
% 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/SUBSET__DEF: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm: h4/pred__set/IN__CROSS: !x Q P. h4/bool/IN x (h4/pred__set/CROSS P Q) <=> h4/bool/IN (h4/pair/FST x) P /\ h4/bool/IN (h4/pair/SND x) Q
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/probability/PROB__INCREASING: !t s p. h4/probability/prob__space p /\ h4/bool/IN s (h4/probability/events p) /\ h4/bool/IN t (h4/probability/events p) /\ h4/pred__set/SUBSET s t ==> h4/real/real__lte (h4/probability/prob p s) (h4/probability/prob p t)
% Assm: h4/probability/distribution__lebesgue__thm1: !s p X A. h4/probability/random__variable X p s /\ h4/bool/IN A (h4/measure/subsets s) ==> h4/extreal/Normal (h4/probability/distribution p X A) = h4/lebesgue/integral p (h4/measure/indicator__fn (h4/pred__set/INTER (h4/util__prob/PREIMAGE X A) (h4/probability/p__space p)))
% Assm: h4/probability/marginal__joint__zero: !t s p Y X. h4/probability/prob__space p /\ h4/probability/events p = h4/pred__set/POW (h4/probability/p__space p) /\ (h4/probability/distribution p X s = h4/real/real__of__num h4/num/0 \/ h4/probability/distribution p Y t = h4/real/real__of__num h4/num/0) ==> h4/probability/joint__distribution p X Y (h4/pred__set/CROSS s t) = h4/real/real__of__num h4/num/0
% Assm: h4/probability/marginal__joint__zero3: !u t s p Z Y X. h4/probability/prob__space p /\ h4/probability/events p = h4/pred__set/POW (h4/probability/p__space p) /\ (h4/probability/distribution p X s = h4/real/real__of__num h4/num/0 \/ h4/probability/distribution p Y t = h4/real/real__of__num h4/num/0 \/ h4/probability/distribution p Z u = h4/real/real__of__num h4/num/0) ==> h4/probability/joint__distribution3 p X Y Z (h4/pred__set/CROSS s (h4/pred__set/CROSS t u)) = h4/real/real__of__num h4/num/0
% Assm: h4/probability/joint__distribution__sym: !p b a Y X. h4/probability/prob__space p ==> h4/probability/joint__distribution p X Y (h4/pred__set/CROSS a b) = h4/probability/joint__distribution p Y X (h4/pred__set/CROSS b a)
% Assm: h4/probability/joint__distribution__pos: !p a Y X. h4/probability/prob__space p /\ h4/probability/events p = h4/pred__set/POW (h4/probability/p__space p) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/probability/joint__distribution p X Y a)
% Assm: h4/probability/joint__distribution__le__1: !p a Y X. h4/probability/prob__space p /\ h4/probability/events p = h4/pred__set/POW (h4/probability/p__space p) ==> h4/real/real__lte (h4/probability/joint__distribution p X Y a) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/probability/PROB__POSITIVE: !s p. h4/probability/prob__space p /\ h4/bool/IN s (h4/probability/events p) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/probability/prob p s)
% Assm: h4/probability/distribution__partition: !p X. h4/probability/prob__space p /\ (!x. h4/bool/IN x (h4/probability/p__space p) ==> h4/bool/IN (h4/pred__set/INSERT x h4/pred__set/EMPTY) (h4/probability/events p)) /\ h4/pred__set/FINITE (h4/probability/p__space p) /\ h4/probability/random__variable X p h4/measure/Borel ==> h4/real__sigma/REAL__SUM__IMAGE (\x. h4/probability/distribution p X (h4/pred__set/INSERT x h4/pred__set/EMPTY)) (h4/pred__set/IMAGE X (h4/probability/p__space p)) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/probability/indep__rv__def: !t s p Y X. h4/probability/indep__rv p X Y s t <=> (!A B. h4/bool/IN A (h4/measure/subsets s) /\ h4/bool/IN B (h4/measure/subsets t) ==> h4/probability/indep p (h4/util__prob/PREIMAGE X A) (h4/util__prob/PREIMAGE Y B))
% Assm: h4/lebesgue/finite__POW__prod__measure__reduce3: !m2 m1 m0. h4/measure/measure__space m0 /\ h4/measure/measure__space m1 /\ h4/measure/measure__space m2 /\ h4/pred__set/FINITE (h4/measure/m__space m0) /\ h4/pred__set/FINITE (h4/measure/m__space m1) /\ h4/pred__set/FINITE (h4/measure/m__space m2) /\ h4/pred__set/POW (h4/measure/m__space m0) = h4/measure/measurable__sets m0 /\ h4/pred__set/POW (h4/measure/m__space m1) = h4/measure/measurable__sets m1 /\ h4/pred__set/POW (h4/measure/m__space m2) = h4/measure/measurable__sets m2 ==> (!a0 a1 a2. h4/bool/IN a0 (h4/measure/measurable__sets m0) /\ h4/bool/IN a1 (h4/measure/measurable__sets m1) /\ h4/bool/IN a2 (h4/measure/measurable__sets m2) ==> h4/lebesgue/prod__measure3 m0 m1 m2 (h4/pred__set/CROSS a0 (h4/pred__set/CROSS a1 a2)) = h4/realax/real__mul (h4/realax/real__mul (h4/measure/measure m0 a0) (h4/measure/measure m1 a1)) (h4/measure/measure m2 a2))
% Assm: h4/probability/distribution__x__eq__1__imp__distribution__y__eq__0: !x p X. h4/probability/random__variable X p (h4/pair/_2C (h4/pred__set/IMAGE X (h4/probability/p__space p)) (h4/pred__set/POW (h4/pred__set/IMAGE X (h4/probability/p__space p)))) /\ h4/probability/distribution p X (h4/pred__set/INSERT x h4/pred__set/EMPTY) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) ==> (!y. ~(y = x) ==> h4/probability/distribution p X (h4/pred__set/INSERT y h4/pred__set/EMPTY) = h4/real/real__of__num h4/num/0)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% 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/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/probability/random__variable__def: !s p X. h4/probability/random__variable X p s <=> h4/probability/prob__space p /\ h4/bool/IN X (h4/measure/measurable (h4/pair/_2C (h4/probability/p__space p) (h4/probability/events p)) s)
% 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/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% 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/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/combin/S__DEF: h4/combin/S = (\f g x. f x (g x))
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/probability/prob__def: h4/probability/prob = h4/measure/measure
% Assm: h4/measure/measure__def: !sts sp mu. h4/measure/measure (h4/pair/_2C sp (h4/pair/_2C sts mu)) = mu
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/probability/p__space__def: h4/probability/p__space = h4/measure/m__space
% Assm: h4/measure/IN__MEASURABLE: !f b a. h4/bool/IN f (h4/measure/measurable a b) <=> h4/measure/sigma__algebra a /\ h4/measure/sigma__algebra b /\ h4/bool/IN f (h4/util__prob/FUNSET (h4/measure/space a) (h4/measure/space b)) /\ (!s. h4/bool/IN s (h4/measure/subsets b) ==> h4/bool/IN (h4/pred__set/INTER (h4/util__prob/PREIMAGE f s) (h4/measure/space a)) (h4/measure/subsets a))
% Assm: h4/probability/events__def: h4/probability/events = h4/measure/measurable__sets
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/measure/measurable__sets__def: !sts sp mu. h4/measure/measurable__sets (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sts
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/probability/PROB__LE__1: !s p. h4/probability/prob__space p /\ h4/bool/IN s (h4/probability/events p) ==> h4/real/real__lte (h4/probability/prob p s) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm: h4/probability/prob__space__def: !p. h4/probability/prob__space p <=> h4/measure/measure__space p /\ h4/measure/measure p (h4/probability/p__space p) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/lebesgue/integral__indicator: !s m. h4/measure/measure__space m /\ h4/bool/IN s (h4/measure/measurable__sets m) ==> h4/lebesgue/integral m (h4/measure/indicator__fn s) = h4/extreal/Normal (h4/measure/measure m s)
% Assm: h4/pred__set/CROSS__DEF: !Q P. h4/pred__set/CROSS P Q = h4/pred__set/GSPEC (\p. h4/pair/_2C p (h4/bool/IN (h4/pair/FST p) P /\ h4/bool/IN (h4/pair/SND p) Q))
% Assm: h4/pred__set/FINITE__CROSS: !Q P. h4/pred__set/FINITE P /\ h4/pred__set/FINITE Q ==> h4/pred__set/FINITE (h4/pred__set/CROSS P Q)
% 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/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/measure/POW__SIGMA__ALGEBRA: !sp. h4/measure/sigma__algebra (h4/pair/_2C sp (h4/pred__set/POW sp))
% Assm: h4/probability/PROB__EMPTY: !p. h4/probability/prob__space p ==> h4/probability/prob p h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/BOUNDED__THM: !v. h4/bool/BOUNDED v <=> T
% Assm: h4/probability/PROB__ADDITIVE: !u t s p. h4/probability/prob__space p /\ h4/bool/IN s (h4/probability/events p) /\ h4/bool/IN t (h4/probability/events p) /\ h4/pred__set/DISJOINT s t /\ u = h4/pred__set/UNION s t ==> h4/probability/prob p u = h4/realax/real__add (h4/probability/prob p s) (h4/probability/prob p t)
% Assm: h4/measure/finite__additivity__sufficient__for__finite__spaces: !s m. h4/measure/sigma__algebra s /\ h4/pred__set/FINITE (h4/measure/space s) /\ h4/measure/positive (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) m)) /\ h4/measure/additive (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) m)) ==> h4/measure/measure__space (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) m))
% Assm: h4/measure/additive__def: !m. h4/measure/additive m <=> (!s t. h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/bool/IN t (h4/measure/measurable__sets m) /\ h4/pred__set/DISJOINT s t ==> h4/measure/measure m (h4/pred__set/UNION s t) = h4/realax/real__add (h4/measure/measure m s) (h4/measure/measure m t))
% Assm: h4/util__prob/PREIMAGE__EMPTY: !f. h4/util__prob/PREIMAGE f h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm: h4/measure/positive__def: !m. h4/measure/positive m <=> h4/measure/measure m h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0 /\ (!s. h4/bool/IN s (h4/measure/measurable__sets m) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/measure/measure m s))
% Assm: h4/pred__set/IN__DISJOINT: !t s. h4/pred__set/DISJOINT s t <=> ~(?x. h4/bool/IN x s /\ h4/bool/IN x t)
% Assm: h4/pred__set/INTER__EMPTY_c0: !s. h4/pred__set/INTER h4/pred__set/EMPTY s = h4/pred__set/EMPTY
% Assm: h4/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x 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/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/pred__set/INTER__SUBSET_c0: !t s. h4/pred__set/SUBSET (h4/pred__set/INTER s t) s
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/real/REAL__LE__ANTISYM: !y x. h4/real/real__lte x y /\ h4/real/real__lte y x <=> x = y
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/probability/joint__distribution3__def: !p Z Y X. h4/probability/joint__distribution3 p X Y Z = (\a. h4/probability/prob p (h4/pred__set/INTER (h4/util__prob/PREIMAGE (\x. h4/pair/_2C (X x) (h4/pair/_2C (Y x) (Z x))) a) (h4/probability/p__space p)))
% Assm: h4/measure/m__space__def: !sts sp mu. h4/measure/m__space (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sp
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/probability/distribution__prob__space: !s p X. h4/probability/random__variable X p s ==> h4/probability/prob__space (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) (h4/probability/distribution p X)))
% Assm: h4/pred__set/IN__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = f x /\ h4/bool/IN x s)
% Assm: h4/pred__set/IN__SING: !y x. h4/bool/IN x (h4/pred__set/INSERT y h4/pred__set/EMPTY) <=> x = y
% Assm: h4/lebesgue/finite__prod__measure__space__POW: !v u s2 s1. h4/pred__set/FINITE s1 /\ h4/pred__set/FINITE s2 ==> h4/lebesgue/prod__measure__space (h4/pair/_2C s1 (h4/pair/_2C (h4/pred__set/POW s1) u)) (h4/pair/_2C s2 (h4/pair/_2C (h4/pred__set/POW s2) v)) = h4/pair/_2C (h4/pred__set/CROSS s1 s2) (h4/pair/_2C (h4/pred__set/POW (h4/pred__set/CROSS s1 s2)) (h4/lebesgue/prod__measure (h4/pair/_2C s1 (h4/pair/_2C (h4/pred__set/POW s1) u)) (h4/pair/_2C s2 (h4/pair/_2C (h4/pred__set/POW s2) v))))
% Assm: h4/lebesgue/measure__space__finite__prod__measure__POW1: !m1 m0. h4/measure/measure__space m0 /\ h4/measure/measure__space m1 /\ h4/pred__set/FINITE (h4/measure/m__space m0) /\ h4/pred__set/FINITE (h4/measure/m__space m1) /\ h4/pred__set/POW (h4/measure/m__space m0) = h4/measure/measurable__sets m0 /\ h4/pred__set/POW (h4/measure/m__space m1) = h4/measure/measurable__sets m1 ==> h4/measure/measure__space (h4/lebesgue/prod__measure__space m0 m1)
% Assm: h4/lebesgue/finite__POW__prod__measure__reduce: !m1 m0. h4/measure/measure__space m0 /\ h4/measure/measure__space m1 /\ h4/pred__set/FINITE (h4/measure/m__space m0) /\ h4/pred__set/FINITE (h4/measure/m__space m1) /\ h4/pred__set/POW (h4/measure/m__space m0) = h4/measure/measurable__sets m0 /\ h4/pred__set/POW (h4/measure/m__space m1) = h4/measure/measurable__sets m1 ==> (!a0 a1. h4/bool/IN a0 (h4/measure/measurable__sets m0) /\ h4/bool/IN a1 (h4/measure/measurable__sets m1) ==> h4/lebesgue/prod__measure m0 m1 (h4/pred__set/CROSS a0 a1) = h4/realax/real__mul (h4/measure/measure m0 a0) (h4/measure/measure m1 a1))
% Assm: h4/measure/MEASURE__SPACE__SUBSET__MSPACE: !m A. h4/measure/measure__space m /\ h4/bool/IN A (h4/measure/measurable__sets m) ==> h4/pred__set/SUBSET A (h4/measure/m__space m)
% Assm: h4/lebesgue/prod__measure__space__def: !m1 m0. h4/lebesgue/prod__measure__space m0 m1 = h4/pair/_2C (h4/pred__set/CROSS (h4/measure/m__space m0) (h4/measure/m__space m1)) (h4/pair/_2C (h4/measure/subsets (h4/measure/sigma (h4/pred__set/CROSS (h4/measure/m__space m0) (h4/measure/m__space m1)) (h4/util__prob/prod__sets (h4/measure/measurable__sets m0) (h4/measure/measurable__sets m1)))) (h4/lebesgue/prod__measure m0 m1))
% Assm: h4/measure/MEASURE__SPACE__REDUCE: !m. h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/measure/measurable__sets m) (h4/measure/measure m)) = m
% Assm: h4/real/REAL__MUL__ASSOC: !z y x. h4/realax/real__mul x (h4/realax/real__mul y z) = h4/realax/real__mul (h4/realax/real__mul x y) z
% Assm: h4/probability/PROB__DISCRETE__EVENTS: !p. h4/probability/prob__space p /\ h4/pred__set/FINITE (h4/probability/p__space p) /\ (!x. h4/bool/IN x (h4/probability/p__space p) ==> h4/bool/IN (h4/pred__set/INSERT x h4/pred__set/EMPTY) (h4/probability/events p)) ==> (!s. h4/pred__set/SUBSET s (h4/probability/p__space p) ==> h4/bool/IN s (h4/probability/events p))
% Assm: h4/probability/PROB__REAL__SUM__IMAGE__SPACE: !p. h4/probability/prob__space p /\ (!x. h4/bool/IN x (h4/probability/p__space p) ==> h4/bool/IN (h4/pred__set/INSERT x h4/pred__set/EMPTY) (h4/probability/events p)) /\ h4/pred__set/FINITE (h4/probability/p__space p) ==> h4/real__sigma/REAL__SUM__IMAGE (\x. h4/probability/prob p (h4/pred__set/INSERT x h4/pred__set/EMPTY)) (h4/probability/p__space p) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% 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/IMAGE__FINITE: !s. h4/pred__set/FINITE s ==> (!f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s))
% Assm: h4/combin/C__DEF: h4/combin/C = (\f x y. f y x)
% Assm: h4/probability/PROB: !c b a. h4/probability/prob (h4/pair/_2C a (h4/pair/_2C b c)) = c
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/probability/EVENTS: !c b a. h4/probability/events (h4/pair/_2C a (h4/pair/_2C b c)) = b
% Assm: h4/probability/prob__x__eq__1__imp__prob__y__eq__0: !x p. h4/probability/prob__space p /\ h4/bool/IN (h4/pred__set/INSERT x h4/pred__set/EMPTY) (h4/probability/events p) /\ h4/probability/prob p (h4/pred__set/INSERT x h4/pred__set/EMPTY) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) ==> (!y. h4/bool/IN (h4/pred__set/INSERT y h4/pred__set/EMPTY) (h4/probability/events p) /\ ~(y = x) ==> h4/probability/prob p (h4/pred__set/INSERT y h4/pred__set/EMPTY) = h4/real/real__of__num h4/num/0)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/real/eq__ints_c0: !n m. h4/real/real__of__num n = h4/real/real__of__num m <=> n = m
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/numeral/numeral__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Goal: !p b a Y X. h4/probability/prob__space p /\ h4/probability/events p = h4/pred__set/POW (h4/probability/p__space p) ==> h4/probability/joint__distribution p X Y (h4/pred__set/CROSS a b) = h4/realax/real__mul (h4/probability/conditional__distribution p Y X b a) (h4/probability/distribution p X a)
%   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_probabilitys_jointu_u_distributionu_u_le]: !p b a Y X. h4/probability/prob__space p /\ happ h4/probability/events p = h4/pred__set/POW (happ h4/probability/p__space p) ==> h4/real/real__lte (happ (h4/probability/joint__distribution p X Y) (h4/pred__set/CROSS a b)) (happ (h4/probability/distribution p X) a)
% Assm [h4s_probabilitys_conditionalu_u_distributionu_u_def]: !p b a Y X. h4/probability/conditional__distribution p X Y a b = h4/real/_2F (happ (h4/probability/joint__distribution p X Y) (h4/pred__set/CROSS a b)) (happ (h4/probability/distribution p Y) b)
% Assm [h4s_probabilitys_jointu_u_distributionu_u_le2]: !p b a Y X. h4/probability/prob__space p /\ happ h4/probability/events p = h4/pred__set/POW (happ h4/probability/p__space p) ==> h4/real/real__lte (happ (h4/probability/joint__distribution p X Y) (h4/pred__set/CROSS a b)) (happ (h4/probability/distribution p Y) b)
% Assm [h4s_probabilitys_distributionu_u_def]: !p X x. happ (h4/probability/distribution p X) x = happ (happ h4/probability/prob p) (h4/pred__set/INTER (h4/util__prob/PREIMAGE X x) (happ h4/probability/p__space p))
% Assm [h4s_probabilitys_finiteu_u_marginalu_u_productu_u_spaceu_u_POW2]: !s2 s1 p Y X. h4/pred__set/POW (happ h4/probability/p__space p) = happ h4/probability/events p /\ h4/probability/random__variable X p (h4/pair/_2C s1 (h4/pred__set/POW s1)) /\ h4/probability/random__variable Y p (h4/pair/_2C s2 (h4/pred__set/POW s2)) /\ h4/pred__set/FINITE (happ h4/probability/p__space p) /\ h4/pred__set/FINITE s1 /\ h4/pred__set/FINITE s2 ==> h4/measure/measure__space (h4/pair/_2C (h4/pred__set/CROSS s1 s2) (h4/pair/_2C (h4/pred__set/POW (h4/pred__set/CROSS s1 s2)) (h4/probability/joint__distribution p X Y)))
% Assm [h4s_probabilitys_jointu_u_distributionu_u_def]: !_0. (!X Y x. happ (happ (happ _0 X) Y) x = h4/pair/_2C (happ X x) (happ Y x)) ==> (!p Y X x. happ (h4/probability/joint__distribution p X Y) x = happ (happ h4/probability/prob p) (h4/pred__set/INTER (h4/util__prob/PREIMAGE (happ (happ _0 X) Y) x) (happ h4/probability/p__space p)))
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_TRUTH]: T
% 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_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_predu_u_sets_INu_u_POW]: !set e. h4/bool/IN e (h4/pred__set/POW set) <=> h4/pred__set/SUBSET e set
% Assm [h4s_predu_u_sets_INTERu_u_SUBSETu_c1]: !t s. h4/pred__set/SUBSET (h4/pred__set/INTER t s) s
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_utilu_u_probs_INu_u_PREIMAGE]: !x s f. h4/bool/IN x (h4/util__prob/PREIMAGE f s) <=> h4/bool/IN (happ f x) s
% 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_SUBSETu_u_DEF]: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_INu_u_CROSS]: !x Q P. h4/bool/IN x (h4/pred__set/CROSS P Q) <=> h4/bool/IN (h4/pair/FST x) P /\ h4/bool/IN (h4/pair/SND x) Q
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_probabilitys_PROBu_u_INCREASING]: !t s p. h4/probability/prob__space p /\ h4/bool/IN s (happ h4/probability/events p) /\ h4/bool/IN t (happ h4/probability/events p) /\ h4/pred__set/SUBSET s t ==> h4/real/real__lte (happ (happ h4/probability/prob p) s) (happ (happ h4/probability/prob p) t)
% Assm [h4s_probabilitys_distributionu_u_lebesgueu_u_thm1]: !s p X A. h4/probability/random__variable X p s /\ h4/bool/IN A (h4/measure/subsets s) ==> h4/extreal/Normal (happ (h4/probability/distribution p X) A) = h4/lebesgue/integral p (h4/measure/indicator__fn (h4/pred__set/INTER (h4/util__prob/PREIMAGE X A) (happ h4/probability/p__space p)))
% Assm [h4s_probabilitys_marginalu_u_jointu_u_zero]: !t s p Y X. h4/probability/prob__space p /\ happ h4/probability/events p = h4/pred__set/POW (happ h4/probability/p__space p) /\ (happ (h4/probability/distribution p X) s = h4/real/real__of__num h4/num/0 \/ happ (h4/probability/distribution p Y) t = h4/real/real__of__num h4/num/0) ==> happ (h4/probability/joint__distribution p X Y) (h4/pred__set/CROSS s t) = h4/real/real__of__num h4/num/0
% Assm [h4s_probabilitys_marginalu_u_jointu_u_zero3]: !u t s p Z Y X. h4/probability/prob__space p /\ happ h4/probability/events p = h4/pred__set/POW (happ h4/probability/p__space p) /\ (happ (h4/probability/distribution p X) s = h4/real/real__of__num h4/num/0 \/ happ (h4/probability/distribution p Y) t = h4/real/real__of__num h4/num/0 \/ happ (h4/probability/distribution p Z) u = h4/real/real__of__num h4/num/0) ==> h4/probability/joint__distribution3 p X Y Z (h4/pred__set/CROSS s (h4/pred__set/CROSS t u)) = h4/real/real__of__num h4/num/0
% Assm [h4s_probabilitys_jointu_u_distributionu_u_sym]: !p b a Y X. h4/probability/prob__space p ==> happ (h4/probability/joint__distribution p X Y) (h4/pred__set/CROSS a b) = happ (h4/probability/joint__distribution p Y X) (h4/pred__set/CROSS b a)
% Assm [h4s_probabilitys_jointu_u_distributionu_u_pos]: !p a Y X. h4/probability/prob__space p /\ happ h4/probability/events p = h4/pred__set/POW (happ h4/probability/p__space p) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ (h4/probability/joint__distribution p X Y) a)
% Assm [h4s_probabilitys_jointu_u_distributionu_u_leu_u_1]: !p a Y X. h4/probability/prob__space p /\ happ h4/probability/events p = h4/pred__set/POW (happ h4/probability/p__space p) ==> h4/real/real__lte (happ (h4/probability/joint__distribution p X Y) a) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_probabilitys_PROBu_u_POSITIVE]: !s p. h4/probability/prob__space p /\ h4/bool/IN s (happ h4/probability/events p) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ (happ h4/probability/prob p) s)
% Assm [h4s_probabilitys_distributionu_u_partition]: !_0. (!p X x. happ (happ (happ _0 p) X) x = happ (h4/probability/distribution p X) (h4/pred__set/INSERT x h4/pred__set/EMPTY)) ==> (!p X. h4/probability/prob__space p /\ (!x. h4/bool/IN x (happ h4/probability/p__space p) ==> h4/bool/IN (h4/pred__set/INSERT x h4/pred__set/EMPTY) (happ h4/probability/events p)) /\ h4/pred__set/FINITE (happ h4/probability/p__space p) /\ h4/probability/random__variable X p h4/measure/Borel ==> h4/real__sigma/REAL__SUM__IMAGE (happ (happ _0 p) X) (h4/pred__set/IMAGE X (happ h4/probability/p__space p)) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_probabilitys_indepu_u_rvu_u_def]: !t s p Y X. h4/probability/indep__rv p X Y s t <=> (!A B. h4/bool/IN A (h4/measure/subsets s) /\ h4/bool/IN B (h4/measure/subsets t) ==> h4/probability/indep p (h4/util__prob/PREIMAGE X A) (h4/util__prob/PREIMAGE Y B))
% Assm [h4s_lebesgues_finiteu_u_POWu_u_produ_u_measureu_u_reduce3]: !m2 m1 m0. h4/measure/measure__space m0 /\ h4/measure/measure__space m1 /\ h4/measure/measure__space m2 /\ h4/pred__set/FINITE (happ h4/measure/m__space m0) /\ h4/pred__set/FINITE (happ h4/measure/m__space m1) /\ h4/pred__set/FINITE (happ h4/measure/m__space m2) /\ h4/pred__set/POW (happ h4/measure/m__space m0) = happ h4/measure/measurable__sets m0 /\ h4/pred__set/POW (happ h4/measure/m__space m1) = happ h4/measure/measurable__sets m1 /\ h4/pred__set/POW (happ h4/measure/m__space m2) = happ h4/measure/measurable__sets m2 ==> (!a0 a1 a2. h4/bool/IN a0 (happ h4/measure/measurable__sets m0) /\ h4/bool/IN a1 (happ h4/measure/measurable__sets m1) /\ h4/bool/IN a2 (happ h4/measure/measurable__sets m2) ==> h4/lebesgue/prod__measure3 m0 m1 m2 (h4/pred__set/CROSS a0 (h4/pred__set/CROSS a1 a2)) = h4/realax/real__mul (h4/realax/real__mul (happ (happ h4/measure/measure m0) a0) (happ (happ h4/measure/measure m1) a1)) (happ (happ h4/measure/measure m2) a2))
% Assm [h4s_probabilitys_distributionu_u_xu_u_equ_u_1u_u_impu_u_distributionu_u_yu_u_equ_u_0]: !x p X. h4/probability/random__variable X p (h4/pair/_2C (h4/pred__set/IMAGE X (happ h4/probability/p__space p)) (h4/pred__set/POW (h4/pred__set/IMAGE X (happ h4/probability/p__space p)))) /\ happ (h4/probability/distribution p X) (h4/pred__set/INSERT x h4/pred__set/EMPTY) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) ==> (!y. ~(y = x) ==> happ (h4/probability/distribution p X) (h4/pred__set/INSERT y h4/pred__set/EMPTY) = h4/real/real__of__num h4/num/0)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% 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_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_probabilitys_randomu_u_variableu_u_def]: !s p X. h4/probability/random__variable X p s <=> h4/probability/prob__space p /\ h4/bool/IN X (h4/measure/measurable (h4/pair/_2C (happ h4/probability/p__space p) (happ h4/probability/events p)) s)
% 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_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_combins_ou_u_DEF]: !g f x. h4/combin/o f g x = happ f (happ g x)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% 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_FALSITY]: !t. F ==> t
% Assm [h4s_probabilitys_probu_u_def]: h4/probability/prob = h4/measure/measure
% Assm [h4s_measures_measureu_u_def]: !sts sp mu. happ h4/measure/measure (h4/pair/_2C sp (h4/pair/_2C sts mu)) = mu
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_probabilitys_pu_u_spaceu_u_def]: h4/probability/p__space = h4/measure/m__space
% Assm [h4s_measures_INu_u_MEASURABLE]: !f b a. h4/bool/IN f (h4/measure/measurable a b) <=> h4/measure/sigma__algebra a /\ h4/measure/sigma__algebra b /\ h4/bool/IN f (h4/util__prob/FUNSET (h4/measure/space a) (h4/measure/space b)) /\ (!s. h4/bool/IN s (h4/measure/subsets b) ==> h4/bool/IN (h4/pred__set/INTER (h4/util__prob/PREIMAGE f s) (h4/measure/space a)) (h4/measure/subsets a))
% Assm [h4s_probabilitys_eventsu_u_def]: h4/probability/events = h4/measure/measurable__sets
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_measures_measurableu_u_setsu_u_def]: !sts sp mu. happ h4/measure/measurable__sets (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sts
% 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_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_probabilitys_PROBu_u_LEu_u_1]: !s p. h4/probability/prob__space p /\ h4/bool/IN s (happ h4/probability/events p) ==> h4/real/real__lte (happ (happ h4/probability/prob p) s) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_probabilitys_probu_u_spaceu_u_def]: !p. h4/probability/prob__space p <=> h4/measure/measure__space p /\ happ (happ h4/measure/measure p) (happ h4/probability/p__space p) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_lebesgues_integralu_u_indicator]: !s m. h4/measure/measure__space m /\ h4/bool/IN s (happ h4/measure/measurable__sets m) ==> h4/lebesgue/integral m (h4/measure/indicator__fn s) = h4/extreal/Normal (happ (happ h4/measure/measure m) s)
% Assm [h4s_predu_u_sets_CROSSu_u_DEF]: !_0. (!P Q p. ?v. (v <=> h4/bool/IN (h4/pair/FST p) P /\ h4/bool/IN (h4/pair/SND p) Q) /\ happ (happ (happ _0 P) Q) p = h4/pair/_2C p v) ==> (!Q P. h4/pred__set/CROSS P Q = h4/pred__set/GSPEC (happ (happ _0 P) Q))
% Assm [h4s_predu_u_sets_FINITEu_u_CROSS]: !Q P. h4/pred__set/FINITE P /\ h4/pred__set/FINITE Q ==> h4/pred__set/FINITE (h4/pred__set/CROSS P Q)
% 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_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_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_measures_POWu_u_SIGMAu_u_ALGEBRA]: !sp. h4/measure/sigma__algebra (h4/pair/_2C sp (h4/pred__set/POW sp))
% Assm [h4s_probabilitys_PROBu_u_EMPTY]: !p. h4/probability/prob__space p ==> happ (happ h4/probability/prob p) h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_BOUNDEDu_u_THM]: !v. h4/bool/BOUNDED v <=> T
% Assm [h4s_probabilitys_PROBu_u_ADDITIVE]: !u t s p. h4/probability/prob__space p /\ h4/bool/IN s (happ h4/probability/events p) /\ h4/bool/IN t (happ h4/probability/events p) /\ h4/pred__set/DISJOINT s t /\ u = h4/pred__set/UNION s t ==> happ (happ h4/probability/prob p) u = h4/realax/real__add (happ (happ h4/probability/prob p) s) (happ (happ h4/probability/prob p) t)
% Assm [h4s_measures_finiteu_u_additivityu_u_sufficientu_u_foru_u_finiteu_u_spaces]: !s m. h4/measure/sigma__algebra s /\ h4/pred__set/FINITE (h4/measure/space s) /\ h4/measure/positive (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) m)) /\ h4/measure/additive (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) m)) ==> h4/measure/measure__space (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) m))
% Assm [h4s_measures_additiveu_u_def]: !m. h4/measure/additive m <=> (!s t. h4/bool/IN s (happ h4/measure/measurable__sets m) /\ h4/bool/IN t (happ h4/measure/measurable__sets m) /\ h4/pred__set/DISJOINT s t ==> happ (happ h4/measure/measure m) (h4/pred__set/UNION s t) = h4/realax/real__add (happ (happ h4/measure/measure m) s) (happ (happ h4/measure/measure m) t))
% Assm [h4s_utilu_u_probs_PREIMAGEu_u_EMPTY]: !f. h4/util__prob/PREIMAGE f h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm [h4s_measures_positiveu_u_def]: !m. h4/measure/positive m <=> happ (happ h4/measure/measure m) h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0 /\ (!s. h4/bool/IN s (happ h4/measure/measurable__sets m) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ (happ h4/measure/measure m) s))
% Assm [h4s_predu_u_sets_INu_u_DISJOINT]: !t s. h4/pred__set/DISJOINT s t <=> ~(?x. h4/bool/IN x s /\ h4/bool/IN x t)
% Assm [h4s_predu_u_sets_INTERu_u_EMPTYu_c0]: !s. h4/pred__set/INTER h4/pred__set/EMPTY s = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x 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_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_predu_u_sets_INTERu_u_SUBSETu_c0]: !t s. h4/pred__set/SUBSET (h4/pred__set/INTER s t) s
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_reals_REALu_u_LEu_u_ANTISYM]: !y x. h4/real/real__lte x y /\ h4/real/real__lte y x <=> x = y
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_probabilitys_jointu_u_distribution3u_u_def]: !_0. (!X Y Z x. happ (happ (happ (happ _0 X) Y) Z) x = h4/pair/_2C (happ X x) (h4/pair/_2C (happ Y x) (happ Z x))) ==> (!p Z Y X x. h4/probability/joint__distribution3 p X Y Z x = happ (happ h4/probability/prob p) (h4/pred__set/INTER (h4/util__prob/PREIMAGE (happ (happ (happ _0 X) Y) Z) x) (happ h4/probability/p__space p)))
% Assm [h4s_measures_mu_u_spaceu_u_def]: !sts sp mu. happ h4/measure/m__space (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sp
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_probabilitys_distributionu_u_probu_u_space]: !s p X. h4/probability/random__variable X p s ==> h4/probability/prob__space (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) (h4/probability/distribution p X)))
% Assm [h4s_predu_u_sets_INu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = happ f x /\ h4/bool/IN x s)
% Assm [h4s_predu_u_sets_INu_u_SING]: !y x. h4/bool/IN x (h4/pred__set/INSERT y h4/pred__set/EMPTY) <=> x = y
% Assm [h4s_lebesgues_finiteu_u_produ_u_measureu_u_spaceu_u_POW]: !v u s2 s1. h4/pred__set/FINITE s1 /\ h4/pred__set/FINITE s2 ==> h4/lebesgue/prod__measure__space (h4/pair/_2C s1 (h4/pair/_2C (h4/pred__set/POW s1) u)) (h4/pair/_2C s2 (h4/pair/_2C (h4/pred__set/POW s2) v)) = h4/pair/_2C (h4/pred__set/CROSS s1 s2) (h4/pair/_2C (h4/pred__set/POW (h4/pred__set/CROSS s1 s2)) (h4/lebesgue/prod__measure (h4/pair/_2C s1 (h4/pair/_2C (h4/pred__set/POW s1) u)) (h4/pair/_2C s2 (h4/pair/_2C (h4/pred__set/POW s2) v))))
% Assm [h4s_lebesgues_measureu_u_spaceu_u_finiteu_u_produ_u_measureu_u_POW1]: !m1 m0. h4/measure/measure__space m0 /\ h4/measure/measure__space m1 /\ h4/pred__set/FINITE (happ h4/measure/m__space m0) /\ h4/pred__set/FINITE (happ h4/measure/m__space m1) /\ h4/pred__set/POW (happ h4/measure/m__space m0) = happ h4/measure/measurable__sets m0 /\ h4/pred__set/POW (happ h4/measure/m__space m1) = happ h4/measure/measurable__sets m1 ==> h4/measure/measure__space (h4/lebesgue/prod__measure__space m0 m1)
% Assm [h4s_lebesgues_finiteu_u_POWu_u_produ_u_measureu_u_reduce]: !m1 m0. h4/measure/measure__space m0 /\ h4/measure/measure__space m1 /\ h4/pred__set/FINITE (happ h4/measure/m__space m0) /\ h4/pred__set/FINITE (happ h4/measure/m__space m1) /\ h4/pred__set/POW (happ h4/measure/m__space m0) = happ h4/measure/measurable__sets m0 /\ h4/pred__set/POW (happ h4/measure/m__space m1) = happ h4/measure/measurable__sets m1 ==> (!a0 a1. h4/bool/IN a0 (happ h4/measure/measurable__sets m0) /\ h4/bool/IN a1 (happ h4/measure/measurable__sets m1) ==> happ (h4/lebesgue/prod__measure m0 m1) (h4/pred__set/CROSS a0 a1) = h4/realax/real__mul (happ (happ h4/measure/measure m0) a0) (happ (happ h4/measure/measure m1) a1))
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_SUBSETu_u_MSPACE]: !m A. h4/measure/measure__space m /\ h4/bool/IN A (happ h4/measure/measurable__sets m) ==> h4/pred__set/SUBSET A (happ h4/measure/m__space m)
% Assm [h4s_lebesgues_produ_u_measureu_u_spaceu_u_def]: !m1 m0. h4/lebesgue/prod__measure__space m0 m1 = h4/pair/_2C (h4/pred__set/CROSS (happ h4/measure/m__space m0) (happ h4/measure/m__space m1)) (h4/pair/_2C (h4/measure/subsets (h4/measure/sigma (h4/pred__set/CROSS (happ h4/measure/m__space m0) (happ h4/measure/m__space m1)) (h4/util__prob/prod__sets (happ h4/measure/measurable__sets m0) (happ h4/measure/measurable__sets m1)))) (h4/lebesgue/prod__measure m0 m1))
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_REDUCE]: !m. h4/pair/_2C (happ h4/measure/m__space m) (h4/pair/_2C (happ h4/measure/measurable__sets m) (happ h4/measure/measure m)) = m
% Assm [h4s_reals_REALu_u_MULu_u_ASSOC]: !z y x. h4/realax/real__mul x (h4/realax/real__mul y z) = h4/realax/real__mul (h4/realax/real__mul x y) z
% Assm [h4s_probabilitys_PROBu_u_DISCRETEu_u_EVENTS]: !p. h4/probability/prob__space p /\ h4/pred__set/FINITE (happ h4/probability/p__space p) /\ (!x. h4/bool/IN x (happ h4/probability/p__space p) ==> h4/bool/IN (h4/pred__set/INSERT x h4/pred__set/EMPTY) (happ h4/probability/events p)) ==> (!s. h4/pred__set/SUBSET s (happ h4/probability/p__space p) ==> h4/bool/IN s (happ h4/probability/events p))
% Assm [h4s_probabilitys_PROBu_u_REALu_u_SUMu_u_IMAGEu_u_SPACE]: !_0. (!p x. happ (happ _0 p) x = happ (happ h4/probability/prob p) (h4/pred__set/INSERT x h4/pred__set/EMPTY)) ==> (!p. h4/probability/prob__space p /\ (!x. h4/bool/IN x (happ h4/probability/p__space p) ==> h4/bool/IN (h4/pred__set/INSERT x h4/pred__set/EMPTY) (happ h4/probability/events p)) /\ h4/pred__set/FINITE (happ h4/probability/p__space p) ==> h4/real__sigma/REAL__SUM__IMAGE (happ _0 p) (happ h4/probability/p__space p) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% 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_IMAGEu_u_FINITE]: !s. h4/pred__set/FINITE s ==> (!f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s))
% Assm [h4s_combins_Cu_u_DEF]: !x x x. h4/combin/C x x x = happ (happ x x) x
% Assm [h4s_probabilitys_PROB]: !c b a. happ h4/probability/prob (h4/pair/_2C a (h4/pair/_2C b c)) = c
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_probabilitys_EVENTS]: !c b a. happ h4/probability/events (h4/pair/_2C a (h4/pair/_2C b c)) = b
% Assm [h4s_probabilitys_probu_u_xu_u_equ_u_1u_u_impu_u_probu_u_yu_u_equ_u_0]: !x p. h4/probability/prob__space p /\ h4/bool/IN (h4/pred__set/INSERT x h4/pred__set/EMPTY) (happ h4/probability/events p) /\ happ (happ h4/probability/prob p) (h4/pred__set/INSERT x h4/pred__set/EMPTY) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) ==> (!y. h4/bool/IN (h4/pred__set/INSERT y h4/pred__set/EMPTY) (happ h4/probability/events p) /\ ~(y = x) ==> happ (happ h4/probability/prob p) (h4/pred__set/INSERT y h4/pred__set/EMPTY) = h4/real/real__of__num h4/num/0)
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_reals_equ_u_intsu_c0]: !n m. h4/real/real__of__num n = h4/real/real__of__num m <=> n = m
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_numerals_numeralu_u_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Goal: !p b a Y X. h4/probability/prob__space p /\ happ h4/probability/events p = h4/pred__set/POW (happ h4/probability/p__space p) ==> happ (h4/probability/joint__distribution p X Y) (h4/pred__set/CROSS a b) = h4/realax/real__mul (h4/probability/conditional__distribution p Y X b a) (happ (h4/probability/distribution p X) a)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1316457,TV_Q1316453]: ![V_f, V_g]: (![V_x]: s(TV_Q1316453,happ(s(t_fun(TV_Q1316457,TV_Q1316453),V_f),s(TV_Q1316457,V_x))) = s(TV_Q1316453,happ(s(t_fun(TV_Q1316457,TV_Q1316453),V_g),s(TV_Q1316457,V_x))) => s(t_fun(TV_Q1316457,TV_Q1316453),V_f) = s(t_fun(TV_Q1316457,TV_Q1316453),V_g))).
fof(ah4s_probabilitys_jointu_u_distributionu_u_le, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_p, V_b, V_a, V_Y, V_X]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_h4s_realaxs_real),h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27b,t_bool),V_a),s(t_fun(TV_u_27c,t_bool),V_b))))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X))),s(t_fun(TV_u_27b,t_bool),V_a)))))))).
fof(ah4s_probabilitys_conditionalu_u_distributionu_u_def, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_p, V_b, V_a, V_Y, V_X]: s(t_h4s_realaxs_real,h4s_probabilitys_conditionalu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y),s(t_fun(TV_u_27b,t_bool),V_a),s(t_fun(TV_u_27c,t_bool),V_b))) = s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_h4s_realaxs_real),h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27b,t_bool),V_a),s(t_fun(TV_u_27c,t_bool),V_b))))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27c),V_Y))),s(t_fun(TV_u_27c,t_bool),V_b)))))).
fof(ah4s_probabilitys_jointu_u_distributionu_u_le2, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_p, V_b, V_a, V_Y, V_X]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_h4s_realaxs_real),h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27b,t_bool),V_a),s(t_fun(TV_u_27c,t_bool),V_b))))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27c),V_Y))),s(t_fun(TV_u_27c,t_bool),V_b)))))))).
fof(ah4s_probabilitys_distributionu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_X, V_x]: s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27b,TV_u_27a),V_X))),s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27b,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27b,TV_u_27a),V_X),s(t_fun(TV_u_27a,t_bool),V_x))),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27b,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_p)))))))).
fof(ah4s_probabilitys_finiteu_u_marginalu_u_productu_u_spaceu_u_POW2, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_s2, V_s1, V_p, V_Y, V_X]: ((s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p))))) = s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))) & (p(s(t_bool,h4s_probabilitys_randomu_u_variable(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_bool),V_s1),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27b,t_bool),V_s1)))))))) & (p(s(t_bool,h4s_probabilitys_randomu_u_variable(s(t_fun(TV_u_27a,TV_u_27c),V_Y),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27c,t_bool),V_s2),s(t_fun(t_fun(TV_u_27c,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27c,t_bool),V_s2)))))))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),V_s1)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27c,t_bool),V_s2))))))))) => p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_h4s_pairs_prod(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27b,t_bool),V_s1),s(t_fun(TV_u_27c,t_bool),V_s2))),s(t_h4s_pairs_prod(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27b,t_bool),V_s1),s(t_fun(TV_u_27c,t_bool),V_s2))))),s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_h4s_realaxs_real),h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y)))))))))))).
fof(ah4s_probabilitys_jointu_u_distributionu_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_X, V_Y, V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_X))),s(t_fun(TV_u_27c,TV_u_27b),V_Y))),s(TV_u_27c,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_X),s(TV_u_27c,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_Y),s(TV_u_27c,V_x))))) => ![V_p, V_Y, V_X, V_x]: s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_realaxs_real),h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27c,TV_u_27a),V_X),s(t_fun(TV_u_27c,TV_u_27b),V_Y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_x))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27c,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_X))),s(t_fun(TV_u_27c,TV_u_27b),V_Y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_x))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27c,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_p))))))))).
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_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_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_predu_u_sets_INu_u_POW, axiom, ![TV_u_27a]: ![V_set, V_e]: s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_e),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,t_bool),V_set))))) = s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_e),s(t_fun(TV_u_27a,t_bool),V_set)))).
fof(ah4s_predu_u_sets_INTERu_u_SUBSETu_c1, axiom, ![TV_u_27a]: ![V_t, V_s]: p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_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_utilu_u_probs_INu_u_PREIMAGE, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_s, V_f]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27b,t_bool),V_s))))) = 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_s)))).
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_SUBSETu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ![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)))) => 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_INu_u_CROSS, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_Q, V_P]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27b,t_bool),V_Q))))))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(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_27a,V_x)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(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_27b,V_y)).
fof(ah4s_probabilitys_PROBu_u_INCREASING, axiom, ![TV_u_27a]: ![V_t, V_s, V_p]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & (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),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & p(s(t_bool,h4s_predu_u_sets_subset(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_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_t)))))))).
fof(ah4s_probabilitys_distributionu_u_lebesgueu_u_thm1, axiom, ![TV_u_27b,TV_u_27a]: ![V_s, V_p, V_X, V_A]: ((p(s(t_bool,h4s_probabilitys_randomu_u_variable(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27b,t_bool),V_A),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s))))))) => s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X))),s(t_fun(TV_u_27b,t_bool),V_A))))) = s(t_h4s_extreals_extreal,h4s_lebesgues_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_p),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_indicatoru_u_fn(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27b,t_bool),V_A))),s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p))))))))))).
fof(ah4s_probabilitys_marginalu_u_jointu_u_zero, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_t, V_s, V_p, V_Y, V_X]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & (s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p))))) & (s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X))),s(t_fun(TV_u_27b,t_bool),V_s))) = 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_fun(TV_u_27c,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27c),V_Y))),s(t_fun(TV_u_27c,t_bool),V_t))) = 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_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_h4s_realaxs_real),h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27b,t_bool),V_s),s(t_fun(TV_u_27c,t_bool),V_t))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_probabilitys_marginalu_u_jointu_u_zero3, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c,TV_u_27d]: ![V_u, V_t, V_s, V_p, V_Z, V_Y, V_X]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & (s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p))))) & (s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X))),s(t_fun(TV_u_27b,t_bool),V_s))) = 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_fun(TV_u_27c,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27c),V_Y))),s(t_fun(TV_u_27c,t_bool),V_t))) = 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_fun(TV_u_27d,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27d),V_Z))),s(t_fun(TV_u_27d,t_bool),V_u))) = 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,h4s_probabilitys_jointu_u_distribution3(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y),s(t_fun(TV_u_27a,TV_u_27d),V_Z),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27c,TV_u_27d)),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27b,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27c,t_bool),V_t),s(t_fun(TV_u_27d,t_bool),V_u))))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_probabilitys_jointu_u_distributionu_u_sym, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_p, V_b, V_a, V_Y, V_X]: (p(s(t_bool,h4s_probabilitys_probu_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_p)))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_h4s_realaxs_real),h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27b,t_bool),V_a),s(t_fun(TV_u_27c,t_bool),V_b))))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),t_bool),t_h4s_realaxs_real),h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27c),V_Y),s(t_fun(TV_u_27a,TV_u_27b),V_X))),s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27c,t_bool),V_b),s(t_fun(TV_u_27b,t_bool),V_a))))))).
fof(ah4s_probabilitys_jointu_u_distributionu_u_pos, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_p, V_a, V_Y, V_X]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))) => 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_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_h4s_realaxs_real),h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),V_a)))))))).
fof(ah4s_probabilitys_jointu_u_distributionu_u_leu_u_1, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_p, V_a, V_Y, V_X]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_h4s_realaxs_real),h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),V_a))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))))).
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_probabilitys_PROBu_u_POSITIVE, axiom, ![TV_u_27a]: ![V_s, V_p]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & 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),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))))) => 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_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_probabilitys_distributionu_u_partition, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_p, V_X, V_x]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_extreals_extreal,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_h4s_realaxs_real)),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_realaxs_real))),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_p))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_X))),s(t_h4s_extreals_extreal,V_x))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_X))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_insert(s(t_h4s_extreals_extreal,V_x),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_empty))))) => ![V_p, V_X]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & (![V_x]: (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(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),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),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))) & p(s(t_bool,h4s_probabilitys_randomu_u_variable(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_X),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),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_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(t_h4s_extreals_extreal,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_extreals_extreal,t_h4s_realaxs_real)),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_realaxs_real))),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_p))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_X))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_X),s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p))))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))).
fof(ah4s_probabilitys_indepu_u_rvu_u_def, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_t, V_s, V_p, V_Y, V_X]: (p(s(t_bool,h4s_probabilitys_indepu_u_rv(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_bool)),V_t)))) <=> ![V_A, V_B]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27b,t_bool),V_A),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s)))))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27c,t_bool),V_B),s(t_fun(t_fun(TV_u_27c,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_bool)),V_t))))))) => p(s(t_bool,h4s_probabilitys_indep(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27b,t_bool),V_A))),s(t_fun(TV_u_27a,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27a,TV_u_27c),V_Y),s(t_fun(TV_u_27c,t_bool),V_B))))))))).
fof(ah4s_lebesgues_finiteu_u_POWu_u_produ_u_measureu_u_reduce3, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_m2, V_m1, V_m0]: ((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_m0)))) & (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1)))) & (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_m2)))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,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(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_m0)))))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27b,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1)))))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27c,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_m2)))))) & (s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,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(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_m0))))) = s(t_fun(t_fun(TV_u_27a,t_bool),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_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_m0))) & (s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27b,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1))))) = s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1))) & s(t_fun(t_fun(TV_u_27c,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27c,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_m2))))) = s(t_fun(t_fun(TV_u_27c,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27c,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_m2))))))))))) => ![V_a0, V_a1, V_a2]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_a0),s(t_fun(t_fun(TV_u_27a,t_bool),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_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_m0)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27b,t_bool),V_a1),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1)))))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27c,t_bool),V_a2),s(t_fun(t_fun(TV_u_27c,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27c,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_m2)))))))) => s(t_h4s_realaxs_real,h4s_lebesgues_produ_u_measure3(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m0),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_m2),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_a0),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27b,t_bool),V_a1),s(t_fun(TV_u_27c,t_bool),V_a2))))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m0))),s(t_fun(TV_u_27a,t_bool),V_a0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1))),s(t_fun(TV_u_27b,t_bool),V_a1))))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_m2))),s(t_fun(TV_u_27c,t_bool),V_a2)))))))).
fof(ah4s_probabilitys_distributionu_u_xu_u_equ_u_1u_u_impu_u_distributionu_u_yu_u_equ_u_0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_p, V_X]: ((p(s(t_bool,h4s_probabilitys_randomu_u_variable(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p))))),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))))))))) & s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,V_x),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))) => ![V_y]: (~ (s(TV_u_27b,V_y) = s(TV_u_27b,V_x)) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
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_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
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,f))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_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_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_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_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_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_probabilitys_randomu_u_variableu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_s, V_p, V_X]: (p(s(t_bool,h4s_probabilitys_randomu_u_variable(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s)))) <=> (p(s(t_bool,h4s_probabilitys_probu_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_p)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(t_fun(TV_u_27a,TV_u_27b),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),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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p))),s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s))))))))).
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_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => 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_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,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_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_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_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_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_probabilitys_probu_u_def, axiom, ![TV_u_27a]: 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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob) = 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_bool),t_h4s_realaxs_real)),h4s_measures_measure)).
fof(ah4s_measures_measureu_u_def, axiom, ![TV_u_27a]: ![V_sts, V_sp, V_mu]: s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_mu))))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_mu)).
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_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_probabilitys_pu_u_spaceu_u_def, axiom, ![TV_u_27a]: 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(TV_u_27a,t_bool)),h4s_probabilitys_pu_u_space) = 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(TV_u_27a,t_bool)),h4s_measures_mu_u_space)).
fof(ah4s_measures_INu_u_MEASURABLE, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_b, V_a]: (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_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(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_b)))))) <=> (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_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_b)))) & (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),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_27b,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_b)))))))) & ![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27b,t_bool),V_s),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_b)))))) => 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_utilu_u_probs_preimage(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27b,t_bool),V_s))),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_probabilitys_eventsu_u_def, axiom, ![TV_u_27a]: 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_bool),t_bool)),h4s_probabilitys_events) = 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_bool),t_bool)),h4s_measures_measurableu_u_sets)).
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_measures_measurableu_u_setsu_u_def, axiom, ![TV_u_27a]: ![V_sts, V_sp, V_mu]: s(t_fun(t_fun(TV_u_27a,t_bool),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_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))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_mu))))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts)).
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_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_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_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_probabilitys_PROBu_u_LEu_u_1, axiom, ![TV_u_27a]: ![V_s, V_p]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & 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),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))))).
fof(ah4s_probabilitys_probu_u_spaceu_u_def, axiom, ![TV_u_27a]: ![V_p]: (p(s(t_bool,h4s_probabilitys_probu_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_p)))) <=> (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) & s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))).
fof(ah4s_lebesgues_integralu_u_indicator, axiom, ![TV_u_27a]: ![V_s, 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_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),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_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_extreals_extreal,h4s_lebesgues_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_indicatoru_u_fn(s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(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),V_s))))))).
fof(ah4s_predu_u_sets_CROSSu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_P, V_Q, V_p]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))),s(t_fun(TV_u_27b,t_bool),V_Q)))))) & s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27b,t_bool),V_Q))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_pairs_u_2c(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p),s(t_bool,V_v)))) => ![V_Q, V_P]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27b,t_bool),V_Q))))))).
fof(ah4s_predu_u_sets_FINITEu_u_CROSS, axiom, ![TV_u_27a,TV_u_27b]: ![V_Q, V_P]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),V_Q))))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))))).
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_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_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_measures_POWu_u_SIGMAu_u_ALGEBRA, axiom, ![TV_u_27a]: ![V_sp]: 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),V_sp),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,t_bool),V_sp))))))))).
fof(ah4s_probabilitys_PROBu_u_EMPTY, axiom, ![TV_u_27a]: ![V_p]: (p(s(t_bool,h4s_probabilitys_probu_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_p)))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_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_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_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_BOUNDEDu_u_THM, axiom, ![V_v]: s(t_bool,h4s_bools_bounded(s(t_bool,V_v))) = s(t_bool,t)).
fof(ah4s_probabilitys_PROBu_u_ADDITIVE, axiom, ![TV_u_27a]: ![V_u, V_t, V_s, V_p]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & (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),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & (p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) & s(t_fun(TV_u_27a,t_bool),V_u) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_u))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_measures_finiteu_u_additivityu_u_sufficientu_u_foru_u_finiteu_u_spaces, axiom, ![TV_u_27a]: ![V_s, 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)),V_s)))) & (p(s(t_bool,h4s_predu_u_sets_finite(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_s)))))) & (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))),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_s))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(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_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_m)))))))) & p(s(t_bool,h4s_measures_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))),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_s))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(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_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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))),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_s))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(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_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_m)))))))))).
fof(ah4s_measures_additiveu_u_def, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_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)))) <=> ![V_s, V_t]: ((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),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_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_bools_in(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(t_fun(TV_u_27a,t_bool),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_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_disjoint(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(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_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(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),V_s))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(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),V_t)))))))).
fof(ah4s_utilu_u_probs_PREIMAGEu_u_EMPTY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: s(t_fun(TV_u_27a,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_measures_positiveu_u_def, axiom, ![TV_u_27a]: ![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)))) <=> (s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(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_empty))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & ![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),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_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_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_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(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),V_s)))))))))).
fof(ah4s_predu_u_sets_INu_u_DISJOINT, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ~ (?[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)))) & 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_INTERu_u_EMPTYu_c0, axiom, ![TV_u_27a]: ![V_s]: 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_empty),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_INu_u_UNION, 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_union(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_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_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_predu_u_sets_INTERu_u_SUBSETu_c0, axiom, ![TV_u_27a]: ![V_t, V_s]: p(s(t_bool,h4s_predu_u_sets_subset(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))),s(t_fun(TV_u_27a,t_bool),V_s))))).
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_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_reals_REALu_u_LEu_u_ANTISYM, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x))))) <=> s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_y))).
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_probabilitys_jointu_u_distribution3u_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c,TV_u_27d]: ![V_uu_0]: (![V_X, V_Y, V_Z, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c))))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27a),t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)))))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27a),V_X))),s(t_fun(TV_u_27d,TV_u_27b),V_Y))),s(t_fun(TV_u_27d,TV_u_27c),V_Z))),s(TV_u_27d,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(TV_u_27d,TV_u_27a),V_X),s(TV_u_27d,V_x))),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_Y),s(TV_u_27d,V_x))),s(TV_u_27c,happ(s(t_fun(TV_u_27d,TV_u_27c),V_Z),s(TV_u_27d,V_x))))))) => ![V_p, V_Z, V_Y, V_X, V_x]: s(t_h4s_realaxs_real,h4s_probabilitys_jointu_u_distribution3(s(t_h4s_pairs_prod(t_fun(TV_u_27d,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27d,t_bool),t_bool),t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27d,TV_u_27a),V_X),s(t_fun(TV_u_27d,TV_u_27b),V_Y),s(t_fun(TV_u_27d,TV_u_27c),V_Z),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),V_x))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27d,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27d,t_bool),t_bool),t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27d,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27d,t_bool),t_bool),t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27d,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27d,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c))))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27a),t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)))))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27a),V_X))),s(t_fun(TV_u_27d,TV_u_27b),V_Y))),s(t_fun(TV_u_27d,TV_u_27c),V_Z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),V_x))),s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27d,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27d,t_bool),t_bool),t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27d,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27d,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27d,t_bool),t_bool),t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real))),V_p))))))))).
fof(ah4s_measures_mu_u_spaceu_u_def, axiom, ![TV_u_27a]: ![V_sts, V_sp, V_mu]: s(t_fun(TV_u_27a,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(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))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_mu))))))) = s(t_fun(TV_u_27a,t_bool),V_sp)).
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_probabilitys_distributionu_u_probu_u_space, axiom, ![TV_u_27a,TV_u_27b]: ![V_s, V_p, V_X]: (p(s(t_bool,h4s_probabilitys_randomu_u_variable(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s)))) => p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s))),s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X)))))))))))).
fof(ah4s_predu_u_sets_INu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> ?[V_x]: (s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_INu_u_SING, axiom, ![TV_u_27a]: ![V_y, V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_lebesgues_finiteu_u_produ_u_measureu_u_spaceu_u_POW, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_u, V_s2, V_s1]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s1)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),V_s2))))) => s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_pairs_prod(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_realaxs_real))),h4s_lebesgues_produ_u_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))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s1),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,t_bool),V_s1))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_u))))),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_bool),V_s2),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27b,t_bool),V_s2))),s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),V_v))))))) = s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_pairs_prod(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_s1),s(t_fun(TV_u_27b,t_bool),V_s2))),s(t_h4s_pairs_prod(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_s1),s(t_fun(TV_u_27b,t_bool),V_s2))))),s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_realaxs_real),h4s_lebesgues_produ_u_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s1),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,t_bool),V_s1))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_u))))),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_bool),V_s2),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27b,t_bool),V_s2))),s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),V_v))))))))))))).
fof(ah4s_lebesgues_measureu_u_spaceu_u_finiteu_u_produ_u_measureu_u_POW1, axiom, ![TV_u_27a,TV_u_27b]: ![V_m1, V_m0]: ((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_m0)))) & (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1)))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,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(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_m0)))))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27b,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1)))))) & (s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,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(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_m0))))) = s(t_fun(t_fun(TV_u_27a,t_bool),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_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_m0))) & s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27b,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1))))) = s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1)))))))) => p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_pairs_prod(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_realaxs_real))),h4s_lebesgues_produ_u_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_m0),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1)))))))).
fof(ah4s_lebesgues_finiteu_u_POWu_u_produ_u_measureu_u_reduce, axiom, ![TV_u_27a,TV_u_27b]: ![V_m1, V_m0]: ((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_m0)))) & (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1)))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,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(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_m0)))))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27b,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1)))))) & (s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,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(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_m0))))) = s(t_fun(t_fun(TV_u_27a,t_bool),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_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_m0))) & s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27b,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1))))) = s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1)))))))) => ![V_a0, V_a1]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_a0),s(t_fun(t_fun(TV_u_27a,t_bool),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_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_m0)))))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27b,t_bool),V_a1),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1))))))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_realaxs_real),h4s_lebesgues_produ_u_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m0),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_a0),s(t_fun(TV_u_27b,t_bool),V_a1))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m0))),s(t_fun(TV_u_27a,t_bool),V_a0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1))),s(t_fun(TV_u_27b,t_bool),V_a1)))))))).
fof(ah4s_measures_MEASUREu_u_SPACEu_u_SUBSETu_u_MSPACE, axiom, ![TV_u_27a]: ![V_m, V_A]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_A),s(t_fun(t_fun(TV_u_27a,t_bool),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_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_subset(s(t_fun(TV_u_27a,t_bool),V_A),s(t_fun(TV_u_27a,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(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_lebesgues_produ_u_measureu_u_spaceu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_m1, V_m0]: s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_pairs_prod(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_realaxs_real))),h4s_lebesgues_produ_u_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_m0),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1))) = s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_pairs_prod(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,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(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_m0))),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27b,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1))))),s(t_h4s_pairs_prod(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool)),h4s_measures_sigma(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,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(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_m0))),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27b,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1))))),s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),h4s_utilu_u_probs_produ_u_sets(s(t_fun(t_fun(TV_u_27a,t_bool),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_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_m0))),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1))))))))),s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_realaxs_real),h4s_lebesgues_produ_u_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m0),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m1)))))))).
fof(ah4s_measures_MEASUREu_u_SPACEu_u_REDUCE, axiom, ![TV_u_27a]: ![V_m]: s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,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(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_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),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_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(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(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(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(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_reals_REALu_u_MULu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,V_z)))).
fof(ah4s_probabilitys_PROBu_u_DISCRETEu_u_EVENTS, axiom, ![TV_u_27a]: ![V_p]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))) & ![V_x]: (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(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),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),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))))))) => ![V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))) => 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),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))))))).
fof(ah4s_probabilitys_PROBu_u_REALu_u_SUMu_u_IMAGEu_u_SPACE, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_p, V_x]: s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),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(TV_u_27a,t_h4s_realaxs_real)),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_p))),s(TV_u_27a,V_x))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))) => ![V_p]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & (![V_x]: (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(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),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),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))))) => s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),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(TV_u_27a,t_h4s_realaxs_real)),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_p))),s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))).
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_IMAGEu_u_FINITE, axiom, ![TV_u_27b,TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_f]: p(s(t_bool,h4s_predu_u_sets_finite(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)))))))).
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_probabilitys_PROB, axiom, ![TV_u_27a]: ![V_c, V_b, V_a]: s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_a),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_b),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_c))))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_c)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_probabilitys_EVENTS, axiom, ![TV_u_27a]: ![V_c, V_b, V_a]: s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_a),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_b),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_c))))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_b)).
fof(ah4s_probabilitys_probu_u_xu_u_equ_u_1u_u_impu_u_probu_u_yu_u_equ_u_0, axiom, ![TV_u_27a]: ![V_x, V_p]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),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),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) => ![V_y]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))),s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & ~ (s(TV_u_27a,V_y) = s(TV_u_27a,V_x))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_reals_equ_u_intsu_c0, axiom, ![V_n, V_m]: (s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,V_m))).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_numerals_numeralu_u_equ_c1, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero) <=> p(s(t_bool,f)))).
fof(ch4s_probabilitys_jointu_u_conditional, conjecture, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_p, V_b, V_a, V_Y, V_X]: ((p(s(t_bool,h4s_probabilitys_probu_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_p)))) & s(t_fun(t_fun(TV_u_27a,t_bool),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_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_pow(s(t_fun(TV_u_27a,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(TV_u_27a,t_bool)),h4s_probabilitys_pu_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_p)))))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),t_h4s_realaxs_real),h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27a,TV_u_27c),V_Y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27b,t_bool),V_a),s(t_fun(TV_u_27c,t_bool),V_b))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_probabilitys_conditionalu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27c),V_Y),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(TV_u_27c,t_bool),V_b),s(t_fun(TV_u_27b,t_bool),V_a))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X))),s(t_fun(TV_u_27b,t_bool),V_a))))))).
