%   ORIGINAL: h4/lebesgue/measure__space__finite__prod__measure__POW3
% 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/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% 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/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% 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/BOUNDED__THM: !v. h4/bool/BOUNDED v <=> T
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% 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/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/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% 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__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/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/DISJOINT__DEF: !t s. h4/pred__set/DISJOINT s t <=> h4/pred__set/INTER s t = h4/pred__set/EMPTY
% 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/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/pred__set/CROSS__EMPTY_c1: !P. h4/pred__set/CROSS h4/pred__set/EMPTY P = h4/pred__set/EMPTY
% 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/IN__POW: !set e. h4/bool/IN e (h4/pred__set/POW set) <=> h4/pred__set/SUBSET e set
% Assm: h4/real/REAL__MUL__LZERO: !x. h4/realax/real__mul (h4/real/real__of__num h4/num/0) x = h4/real/real__of__num h4/num/0
% Assm: h4/real/REAL__LE__MUL: !y x. h4/real/real__lte (h4/real/real__of__num h4/num/0) x /\ h4/real/real__lte (h4/real/real__of__num h4/num/0) y ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/realax/real__mul x y)
% Assm: h4/real/REAL__ADD__RDISTRIB: !z y x. h4/realax/real__mul (h4/realax/real__add x y) z = h4/realax/real__add (h4/realax/real__mul x z) (h4/realax/real__mul y z)
% Assm: h4/real__sigma/REAL__SUM__IMAGE__POS: !s f. h4/pred__set/FINITE s /\ (!x. h4/bool/IN x s ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (f x)) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/real__sigma/REAL__SUM__IMAGE f s)
% Assm: h4/real__sigma/REAL__SUM__IMAGE__ADD: !s. h4/pred__set/FINITE s ==> (!f f_27. h4/real__sigma/REAL__SUM__IMAGE (\x. h4/realax/real__add (f x) (f_27 x)) s = h4/realax/real__add (h4/real__sigma/REAL__SUM__IMAGE f s) (h4/real__sigma/REAL__SUM__IMAGE f_27 s))
% Assm: h4/real__sigma/REAL__SUM__IMAGE__EQ: !s f_27 f. h4/pred__set/FINITE s /\ (!x. h4/bool/IN x s ==> f x = f_27 x) ==> h4/real__sigma/REAL__SUM__IMAGE f s = h4/real__sigma/REAL__SUM__IMAGE f_27 s
% 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/extreal/extreal__distinct_c1: !a. ~(h4/extreal/NegInf = h4/extreal/Normal a)
% Assm: h4/extreal/extreal__distinct_c2: !a. ~(h4/extreal/PosInf = h4/extreal/Normal a)
% Assm: h4/extreal/real__normal: !x. h4/extreal/real (h4/extreal/Normal x) = x
% Assm: h4/extreal/extreal__mul__def_c8: !y x. h4/extreal/extreal__mul (h4/extreal/Normal x) (h4/extreal/Normal y) = h4/extreal/Normal (h4/realax/real__mul x y)
% Assm: h4/extreal/EXTREAL__SUM__IMAGE__NORMAL: !s f. h4/pred__set/FINITE s ==> h4/extreal/EXTREAL__SUM__IMAGE (\x. h4/extreal/Normal (f x)) s = h4/extreal/Normal (h4/real__sigma/REAL__SUM__IMAGE f s)
% Assm: h4/measure/space__def: !y x. h4/measure/space (h4/pair/_2C x y) = x
% Assm: h4/measure/subsets__def: !y x. h4/measure/subsets (h4/pair/_2C x y) = y
% 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/measure/measure__def: !sts sp mu. h4/measure/measure (h4/pair/_2C sp (h4/pair/_2C sts mu)) = mu
% 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/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/measure/POW__SIGMA__ALGEBRA: !sp. h4/measure/sigma__algebra (h4/pair/_2C sp (h4/pred__set/POW sp))
% Assm: h4/measure/MEASURE__EMPTY: !m. h4/measure/measure__space m ==> h4/measure/measure m h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0
% Assm: h4/measure/MEASURE__ADDITIVE: !u t s m. h4/measure/measure__space m /\ h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/bool/IN t (h4/measure/measurable__sets m) /\ h4/pred__set/DISJOINT s t /\ u = h4/pred__set/UNION s t ==> h4/measure/measure m u = h4/realax/real__add (h4/measure/measure m s) (h4/measure/measure m t)
% 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/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/MEASURE__SPACE__POSITIVE: !m. h4/measure/measure__space m ==> h4/measure/positive m
% Assm: h4/measure/MEASURE__SPACE__EMPTY__MEASURABLE: !m. h4/measure/measure__space m ==> h4/bool/IN h4/pred__set/EMPTY (h4/measure/measurable__sets m)
% Assm: h4/lebesgue/prod__measure__def: !m1 m0. h4/lebesgue/prod__measure m0 m1 = (\a. h4/extreal/real (h4/lebesgue/integral m0 (\s0. h4/extreal/Normal (h4/measure/measure m1 (h4/util__prob/PREIMAGE (\s1. h4/pair/_2C s0 s1) a)))))
% Assm: h4/lebesgue/prod__measure3__def: !m2 m1 m0. h4/lebesgue/prod__measure3 m0 m1 m2 = h4/lebesgue/prod__measure m0 (h4/lebesgue/prod__measure__space m1 m2)
% Assm: h4/lebesgue/finite__space__POW__integral__reduce: !m f. h4/measure/measure__space m /\ h4/pred__set/POW (h4/measure/m__space m) = h4/measure/measurable__sets m /\ h4/pred__set/FINITE (h4/measure/m__space m) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> ~(f x = h4/extreal/NegInf) /\ ~(f x = h4/extreal/PosInf)) ==> h4/lebesgue/integral m f = h4/extreal/EXTREAL__SUM__IMAGE (\x. h4/extreal/extreal__mul (f x) (h4/extreal/Normal (h4/measure/measure m (h4/pred__set/INSERT x h4/pred__set/EMPTY)))) (h4/measure/m__space m)
% 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__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/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))
% Goal: !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 ==> h4/measure/measure__space (h4/pair/_2C (h4/pred__set/CROSS (h4/measure/m__space m0) (h4/pred__set/CROSS (h4/measure/m__space m1) (h4/measure/m__space m2))) (h4/pair/_2C (h4/pred__set/POW (h4/pred__set/CROSS (h4/measure/m__space m0) (h4/pred__set/CROSS (h4/measure/m__space m1) (h4/measure/m__space m2)))) (h4/lebesgue/prod__measure3 m0 m1 m2)))
%   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_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% 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_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% 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_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% 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_BOUNDEDu_u_THM]: !v. h4/bool/BOUNDED v <=> T
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~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_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_combins_Iu_u_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% 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_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_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f 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_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% 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_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_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_DISJOINTu_u_DEF]: !t s. h4/pred__set/DISJOINT s t <=> h4/pred__set/INTER s t = h4/pred__set/EMPTY
% 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_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_predu_u_sets_CROSSu_u_EMPTYu_c1]: !P. h4/pred__set/CROSS h4/pred__set/EMPTY P = h4/pred__set/EMPTY
% 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_INu_u_POW]: !set e. h4/bool/IN e (h4/pred__set/POW set) <=> h4/pred__set/SUBSET e set
% Assm [h4s_reals_REALu_u_MULu_u_LZERO]: !x. h4/realax/real__mul (h4/real/real__of__num h4/num/0) x = h4/real/real__of__num h4/num/0
% Assm [h4s_reals_REALu_u_LEu_u_MUL]: !y x. h4/real/real__lte (h4/real/real__of__num h4/num/0) x /\ h4/real/real__lte (h4/real/real__of__num h4/num/0) y ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/realax/real__mul x y)
% Assm [h4s_reals_REALu_u_ADDu_u_RDISTRIB]: !z y x. h4/realax/real__mul (h4/realax/real__add x y) z = h4/realax/real__add (h4/realax/real__mul x z) (h4/realax/real__mul y z)
% Assm [h4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_POS]: !s f. h4/pred__set/FINITE s /\ (!x. h4/bool/IN x s ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ f x)) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/real__sigma/REAL__SUM__IMAGE f s)
% Assm [h4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_ADD]: !_0. (!f f_27 x. happ (happ (happ _0 f) f_27) x = h4/realax/real__add (happ f x) (happ f_27 x)) ==> (!s. h4/pred__set/FINITE s ==> (!f f_27. h4/real__sigma/REAL__SUM__IMAGE (happ (happ _0 f) f_27) s = h4/realax/real__add (h4/real__sigma/REAL__SUM__IMAGE f s) (h4/real__sigma/REAL__SUM__IMAGE f_27 s)))
% Assm [h4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_EQ]: !s f_27 f. h4/pred__set/FINITE s /\ (!x. h4/bool/IN x s ==> happ f x = happ f_27 x) ==> h4/real__sigma/REAL__SUM__IMAGE f s = h4/real__sigma/REAL__SUM__IMAGE f_27 s
% 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_extreals_extrealu_u_distinctu_c1]: !a. ~(h4/extreal/NegInf = h4/extreal/Normal a)
% Assm [h4s_extreals_extrealu_u_distinctu_c2]: !a. ~(h4/extreal/PosInf = h4/extreal/Normal a)
% Assm [h4s_extreals_realu_u_normal]: !x. h4/extreal/real (h4/extreal/Normal x) = x
% Assm [h4s_extreals_extrealu_u_mulu_u_defu_c8]: !y x. h4/extreal/extreal__mul (h4/extreal/Normal x) (h4/extreal/Normal y) = h4/extreal/Normal (h4/realax/real__mul x y)
% Assm [h4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_NORMAL]: !_0. (!f x. happ (happ _0 f) x = h4/extreal/Normal (happ f x)) ==> (!s f. h4/pred__set/FINITE s ==> h4/extreal/EXTREAL__SUM__IMAGE (happ _0 f) s = h4/extreal/Normal (h4/real__sigma/REAL__SUM__IMAGE f s))
% Assm [h4s_measures_spaceu_u_def]: !y x. h4/measure/space (h4/pair/_2C x y) = x
% Assm [h4s_measures_subsetsu_u_def]: !y x. h4/measure/subsets (h4/pair/_2C x y) = y
% Assm [h4s_measures_measurableu_u_setsu_u_def]: !sts sp mu. h4/measure/measurable__sets (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sts
% Assm [h4s_measures_measureu_u_def]: !sts sp mu. h4/measure/measure (h4/pair/_2C sp (h4/pair/_2C sts mu)) = mu
% Assm [h4s_measures_positiveu_u_def]: !m. h4/measure/positive m <=> happ (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) (happ (h4/measure/measure m) s))
% Assm [h4s_measures_additiveu_u_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 ==> happ (h4/measure/measure m) (h4/pred__set/UNION s t) = h4/realax/real__add (happ (h4/measure/measure m) s) (happ (h4/measure/measure m) t))
% Assm [h4s_measures_POWu_u_SIGMAu_u_ALGEBRA]: !sp. h4/measure/sigma__algebra (h4/pair/_2C sp (h4/pred__set/POW sp))
% Assm [h4s_measures_MEASUREu_u_EMPTY]: !m. h4/measure/measure__space m ==> happ (h4/measure/measure m) h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0
% Assm [h4s_measures_MEASUREu_u_ADDITIVE]: !u t s m. h4/measure/measure__space m /\ h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/bool/IN t (h4/measure/measurable__sets m) /\ h4/pred__set/DISJOINT s t /\ u = h4/pred__set/UNION s t ==> happ (h4/measure/measure m) u = h4/realax/real__add (happ (h4/measure/measure m) s) (happ (h4/measure/measure m) t)
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_REDUCE]: !m. h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/measure/measurable__sets m) (h4/measure/measure m)) = m
% 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_MEASUREu_u_SPACEu_u_POSITIVE]: !m. h4/measure/measure__space m ==> h4/measure/positive m
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_EMPTYu_u_MEASURABLE]: !m. h4/measure/measure__space m ==> h4/bool/IN h4/pred__set/EMPTY (h4/measure/measurable__sets m)
% Assm [h4s_lebesgues_produ_u_measureu_u_def]: !_1. (!s0 s1. happ (happ _1 s0) s1 = h4/pair/_2C s0 s1) ==> (!_0. (!m1 x s0. happ (happ (happ _0 m1) x) s0 = h4/extreal/Normal (happ (h4/measure/measure m1) (h4/util__prob/PREIMAGE (happ _1 s0) x))) ==> (!m1 m0 x. happ (h4/lebesgue/prod__measure m0 m1) x = h4/extreal/real (h4/lebesgue/integral m0 (happ (happ _0 m1) x))))
% Assm [h4s_lebesgues_produ_u_measure3u_u_def]: !m2 m1 m0. h4/lebesgue/prod__measure3 m0 m1 m2 = h4/lebesgue/prod__measure m0 (h4/lebesgue/prod__measure__space m1 m2)
% Assm [h4s_lebesgues_finiteu_u_spaceu_u_POWu_u_integralu_u_reduce]: !_0. (!f m x. happ (happ (happ _0 f) m) x = h4/extreal/extreal__mul (happ f x) (h4/extreal/Normal (happ (h4/measure/measure m) (h4/pred__set/INSERT x h4/pred__set/EMPTY)))) ==> (!m f. h4/measure/measure__space m /\ h4/pred__set/POW (h4/measure/m__space m) = h4/measure/measurable__sets m /\ h4/pred__set/FINITE (h4/measure/m__space m) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> ~(happ f x = h4/extreal/NegInf) /\ ~(happ f x = h4/extreal/PosInf)) ==> h4/lebesgue/integral m f = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ _0 f) m) (h4/measure/m__space m))
% 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 (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 [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_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 (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) ==> happ (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 (h4/measure/measure m0) a0) (happ (h4/measure/measure m1) a1)) (happ (h4/measure/measure m2) a2))
% Goal: !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 ==> h4/measure/measure__space (h4/pair/_2C (h4/pred__set/CROSS (h4/measure/m__space m0) (h4/pred__set/CROSS (h4/measure/m__space m1) (h4/measure/m__space m2))) (h4/pair/_2C (h4/pred__set/POW (h4/pred__set/CROSS (h4/measure/m__space m0) (h4/pred__set/CROSS (h4/measure/m__space m1) (h4/measure/m__space m2)))) (h4/lebesgue/prod__measure3 m0 m1 m2)))
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_Q269518,TV_Q269514]: ![V_f, V_g]: (![V_x]: s(TV_Q269514,happ(s(t_fun(TV_Q269518,TV_Q269514),V_f),s(TV_Q269518,V_x))) = s(TV_Q269514,happ(s(t_fun(TV_Q269518,TV_Q269514),V_g),s(TV_Q269518,V_x))) => s(t_fun(TV_Q269518,TV_Q269514),V_f) = s(t_fun(TV_Q269518,TV_Q269514),V_g))).
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_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_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_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_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_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_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_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_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
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_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_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_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_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_BOUNDEDu_u_THM, axiom, ![V_v]: s(t_bool,h4s_bools_bounded(s(t_bool,V_v))) = s(t_bool,t)).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
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_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_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_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_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_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_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_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_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_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_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_Iu_u_ou_u_IDu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_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_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_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_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_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_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_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_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_DISJOINTu_u_DEF, 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)))) <=> 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),h4s_predu_u_sets_empty))).
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_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_predu_u_sets_CROSSu_u_EMPTYu_c1, axiom, ![TV_u_27c,TV_u_27a]: ![V_P]: s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_empty),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27a),t_bool),h4s_predu_u_sets_empty)).
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_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_reals_REALu_u_MULu_u_LZERO, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_reals_REALu_u_LEu_u_MUL, axiom, ![V_y, V_x]: ((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,V_x)))) & 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,V_y))))) => 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,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))))).
fof(ah4s_reals_REALu_u_ADDu_u_RDISTRIB, axiom, ![V_z, V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,V_z))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z)))))).
fof(ah4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_POS, axiom, ![TV_u_27a]: ![V_s, V_f]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => 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(TV_u_27a,t_h4s_realaxs_real),V_f),s(TV_u_27a,V_x)))))))) => 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,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_ADD, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_fu_27, V_x]: s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_fu_27))),s(TV_u_27a,V_x))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(TV_u_27a,V_x))),s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_fu_27),s(TV_u_27a,V_x))))) => ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_f, V_fu_27]: 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_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_fu_27))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_fu_27),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_EQ, axiom, ![TV_u_27a]: ![V_s, V_fu_27, V_f]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(TV_u_27a,V_x))) = s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_fu_27),s(TV_u_27a,V_x))))) => s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_fu_27),s(t_fun(TV_u_27a,t_bool),V_s))))).
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_extreals_extrealu_u_distinctu_c1, axiom, ![V_a]: ~ (s(t_h4s_extreals_extreal,h4s_extreals_neginf) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_a))))).
fof(ah4s_extreals_extrealu_u_distinctu_c2, axiom, ![V_a]: ~ (s(t_h4s_extreals_extreal,h4s_extreals_posinf) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_a))))).
fof(ah4s_extreals_realu_u_normal, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_extreals_real(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_x))))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_extreals_extrealu_u_mulu_u_defu_c8, axiom, ![V_y, V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_x))),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_y))))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_NORMAL, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(TV_u_27a,V_x))))) => ![V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
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_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_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),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_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),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_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),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),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),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_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),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),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),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),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),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_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_measures_MEASUREu_u_EMPTY, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) => s(t_h4s_realaxs_real,happ(s(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))))).
fof(ah4s_measures_MEASUREu_u_ADDITIVE, axiom, ![TV_u_27a]: ![V_u, V_t, 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),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),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_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),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_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),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),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_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),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),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),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_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_MEASUREu_u_SPACEu_u_POSITIVE, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) => p(s(t_bool,h4s_measures_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)))))).
fof(ah4s_measures_MEASUREu_u_SPACEu_u_EMPTYu_u_MEASURABLE, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))).
fof(ah4s_lebesgues_produ_u_measureu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_s0, V_s1]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_uu_1),s(TV_u_27a,V_s0))),s(TV_u_27b,V_s1))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_s0),s(TV_u_27b,V_s1))) => ![V_uu_0]: (![V_m1, V_x, V_s0]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal)),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(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),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),V_x))),s(TV_u_27a,V_s0))) = 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_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),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_uu_1),s(TV_u_27a,V_s0))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_x))))))) => ![V_m1, V_m0, 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_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),V_x))) = s(t_h4s_realaxs_real,h4s_extreals_real(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_m0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal)),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(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),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),V_x)))))))))).
fof(ah4s_lebesgues_produ_u_measure3u_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_m2, V_m1, V_m0]: s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),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_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),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(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_lebesgues_produ_u_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),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)))))).
fof(ah4s_lebesgues_finiteu_u_spaceu_u_POWu_u_integralu_u_reduce, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_m, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),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_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(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(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(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_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))))))) => ![V_m, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (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),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => (~ (s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_neginf)) & ~ (s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf))))))) => 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),V_f))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),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_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(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_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_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),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),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),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),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),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),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_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_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),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),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),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),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),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),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),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),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),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),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),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),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,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),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),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),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),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(ch4s_lebesgues_measureu_u_spaceu_u_finiteu_u_produ_u_measureu_u_POW3, conjecture, ![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),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),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),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),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),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),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),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),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),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))))))))))) => p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,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_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,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_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),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_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),h4s_predu_u_sets_cross(s(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(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_h4s_pairs_prod(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,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_27a,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_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),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_h4s_pairs_prod(TV_u_27b,TV_u_27c),t_bool),h4s_predu_u_sets_cross(s(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(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(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),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)))))))))))).
