%   ORIGINAL: h4/lebesgue/finite__prod__measure__space__POW
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__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/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_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% 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/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% 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/S__DEF: h4/combin/S = (\f g x. f x (g x))
% Assm: h4/combin/C__DEF: h4/combin/C = (\f x y. f y x)
% 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/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% 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/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% 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/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% 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__SING: !y x. h4/bool/IN x (h4/pred__set/INSERT y h4/pred__set/EMPTY) <=> x = y
% 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/SUBSET__FINITE: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/SUBSET t s ==> h4/pred__set/FINITE t)
% 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/pred__set/IN__BIGINTER: !x B. h4/bool/IN x (h4/pred__set/BIGINTER B) <=> (!P. h4/bool/IN P B ==> h4/bool/IN x P)
% 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/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/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/util__prob/FINITE__COUNTABLE: !s. h4/pred__set/FINITE s ==> h4/util__prob/countable s
% Assm: h4/util__prob/prod__sets__def: !b a. h4/util__prob/prod__sets a b = h4/pred__set/GSPEC (h4/pair/UNCURRY (\s t. h4/pair/_2C (h4/pred__set/CROSS s t) (h4/bool/IN s a /\ h4/bool/IN t b)))
% Assm: h4/util__prob/IN__BIGUNION__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f s)) <=> (?x. h4/bool/IN x s /\ h4/bool/IN y (f x))
% Assm: h4/measure/subsets__def: !y x. h4/measure/subsets (h4/pair/_2C x y) = y
% Assm: h4/measure/sigma__def: !st sp. h4/measure/sigma sp st = h4/pair/_2C sp (h4/pred__set/BIGINTER (h4/pred__set/GSPEC (\s. h4/pair/_2C s (h4/pred__set/SUBSET st s /\ h4/measure/sigma__algebra (h4/pair/_2C sp s)))))
% 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/measure/measurable__sets__def: !sts sp mu. h4/measure/measurable__sets (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sts
% Assm: h4/measure/POW__SIGMA__ALGEBRA: !sp. h4/measure/sigma__algebra (h4/pair/_2C sp (h4/pred__set/POW sp))
% Assm: h4/measure/SIGMA__ALGEBRA: !p. h4/measure/sigma__algebra p <=> h4/measure/subset__class (h4/measure/space p) (h4/measure/subsets p) /\ h4/bool/IN h4/pred__set/EMPTY (h4/measure/subsets p) /\ (!s. h4/bool/IN s (h4/measure/subsets p) ==> h4/bool/IN (h4/pred__set/DIFF (h4/measure/space p) s) (h4/measure/subsets p)) /\ (!c. h4/util__prob/countable c /\ h4/pred__set/SUBSET c (h4/measure/subsets p) ==> h4/bool/IN (h4/pred__set/BIGUNION c) (h4/measure/subsets p))
% 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))
% Goal: !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))))
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_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_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_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> 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_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_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_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% 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_Su_u_DEF]: !x x x. h4/combin/S x x x = happ (happ x x) (happ x x)
% Assm [h4s_combins_Cu_u_DEF]: !x x x. h4/combin/C x x x = happ (happ x x) x
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_combins_Iu_u_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_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% 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_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_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_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_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_SUBSETu_u_FINITE]: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/SUBSET t s ==> h4/pred__set/FINITE t)
% 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_predu_u_sets_INu_u_BIGINTER]: !x B. h4/bool/IN x (h4/pred__set/BIGINTER B) <=> (!P. h4/bool/IN P B ==> h4/bool/IN x P)
% 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_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_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_utilu_u_probs_FINITEu_u_COUNTABLE]: !s. h4/pred__set/FINITE s ==> h4/util__prob/countable s
% Assm [h4s_utilu_u_probs_produ_u_setsu_u_def]: !_1. (!s a b t. ?v. (v <=> h4/bool/IN s a /\ h4/bool/IN t b) /\ happ (happ (happ (happ _1 s) a) b) t = h4/pair/_2C (h4/pred__set/CROSS s t) v) ==> (!_0. (!a b s. happ (happ (happ _0 a) b) s = happ (happ (happ _1 s) a) b) ==> (!b a. h4/util__prob/prod__sets a b = h4/pred__set/GSPEC (h4/pair/UNCURRY (happ (happ _0 a) b))))
% Assm [h4s_utilu_u_probs_INu_u_BIGUNIONu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f s)) <=> (?x. h4/bool/IN x s /\ h4/bool/IN y (happ f x))
% Assm [h4s_measures_subsetsu_u_def]: !y x. h4/measure/subsets (h4/pair/_2C x y) = y
% Assm [h4s_measures_sigmau_u_def]: !_0. (!st sp s. ?v. (v <=> h4/pred__set/SUBSET st s /\ h4/measure/sigma__algebra (h4/pair/_2C sp s)) /\ happ (happ (happ _0 st) sp) s = h4/pair/_2C s v) ==> (!st sp. h4/measure/sigma sp st = h4/pair/_2C sp (h4/pred__set/BIGINTER (h4/pred__set/GSPEC (happ (happ _0 st) sp))))
% Assm [h4s_measures_mu_u_spaceu_u_def]: !sts sp mu. h4/measure/m__space (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sp
% 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_POWu_u_SIGMAu_u_ALGEBRA]: !sp. h4/measure/sigma__algebra (h4/pair/_2C sp (h4/pred__set/POW sp))
% Assm [h4s_measures_SIGMAu_u_ALGEBRA]: !p. h4/measure/sigma__algebra p <=> h4/measure/subset__class (h4/measure/space p) (h4/measure/subsets p) /\ h4/bool/IN h4/pred__set/EMPTY (h4/measure/subsets p) /\ (!s. h4/bool/IN s (h4/measure/subsets p) ==> h4/bool/IN (h4/pred__set/DIFF (h4/measure/space p) s) (h4/measure/subsets p)) /\ (!c. h4/util__prob/countable c /\ h4/pred__set/SUBSET c (h4/measure/subsets p) ==> h4/bool/IN (h4/pred__set/BIGUNION c) (h4/measure/subsets p))
% 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 (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))
% Goal: !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))))
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_Q269464,TV_Q269460]: ![V_f, V_g]: (![V_x]: s(TV_Q269460,happ(s(t_fun(TV_Q269464,TV_Q269460),V_f),s(TV_Q269464,V_x))) = s(TV_Q269460,happ(s(t_fun(TV_Q269464,TV_Q269460),V_g),s(TV_Q269464,V_x))) => s(t_fun(TV_Q269464,TV_Q269460),V_f) = s(t_fun(TV_Q269464,TV_Q269460),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_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_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_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_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_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_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_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_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_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_combins_Cu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_x, V_x0, V_x1]: s(TV_u_27c,h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27b,V_x0),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,V_x0)))).
fof(ah4s_combins_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_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_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_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_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_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
fof(ah4s_predu_u_sets_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_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_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_SUBSETu_u_FINITE, axiom, ![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_t]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))))).
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_predu_u_sets_INu_u_BIGINTER, axiom, ![TV_u_27a]: ![V_x, V_B]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_biginter(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_B)))))) <=> ![V_P]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_B)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))))))).
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_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_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_utilu_u_probs_FINITEu_u_COUNTABLE, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_utilu_u_probs_countable(s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_utilu_u_probs_produ_u_setsu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_s, V_a, V_b, V_t]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (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),V_a)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27b,t_bool),V_t),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),V_b)))))) & s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_a))),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),V_b))),s(t_fun(TV_u_27b,t_bool),V_t))) = s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),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_s),s(t_fun(TV_u_27b,t_bool),V_t))),s(t_bool,V_v)))) => ![V_uu_0]: (![V_a, V_b, V_s]: s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool))),happ(s(t_fun(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool)))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool))))),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_a))),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),V_b))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_a))),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),V_b))) => ![V_b, V_a]: 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),V_a),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),V_b))) = s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool)),h4s_pairs_uncurry(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool))),happ(s(t_fun(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool)))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool))))),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_a))),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),V_b)))))))))).
fof(ah4s_utilu_u_probs_INu_u_BIGUNIONu_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_bigunion(s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f),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_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f),s(TV_u_27a,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_sigmau_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_st, V_sp, V_s]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_st),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s)))) & p(s(t_bool,h4s_measures_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),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),V_s)))))))) & s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)))),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_st))),s(t_fun(TV_u_27a,t_bool),V_sp))),s(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_bool),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_s),s(t_bool,V_v)))) => ![V_st, V_sp]: s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_sigma(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_st))) = 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_biginter(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_bool)))),V_uu_0),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_st))),s(t_fun(TV_u_27a,t_bool),V_sp))))))))))).
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),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_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_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_SIGMAu_u_ALGEBRA, axiom, ![TV_u_27a]: ![V_p]: (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_p)))) <=> (p(s(t_bool,h4s_measures_subsetu_u_class(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_p))),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_p)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_p)))))) & (![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_p)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_p))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_p))))))) & ![V_c]: ((p(s(t_bool,h4s_utilu_u_probs_countable(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_c)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_c),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_p))))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_c))),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_p)))))))))))).
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),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),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),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),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),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_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(ch4s_lebesgues_finiteu_u_produ_u_measureu_u_spaceu_u_POW, conjecture, ![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))))))))))))).
