%   ORIGINAL: h4/probability/uniform__distribution__prob__space
% 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/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/OR__CLAUSES_c3: !t. t \/ F <=> 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_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% 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/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% 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/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/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/numeral/numeral__distrib_c18: !n. h4/num/0 = h4/arithmetic/NUMERAL n <=> n = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% 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/CHOICE__DEF: !s. ~(s = h4/pred__set/EMPTY) ==> h4/bool/IN (h4/pred__set/CHOICE s) 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/CARD__EMPTY: h4/pred__set/CARD h4/pred__set/EMPTY = h4/num/0
% Assm: h4/pred__set/CARD__EQ__0: !s. h4/pred__set/FINITE s ==> (h4/pred__set/CARD s = h4/num/0 <=> s = h4/pred__set/EMPTY)
% Assm: h4/pred__set/CARD__UNION: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/FINITE t ==> h4/arithmetic/_2B (h4/pred__set/CARD (h4/pred__set/UNION s t)) (h4/pred__set/CARD (h4/pred__set/INTER s t)) = h4/arithmetic/_2B (h4/pred__set/CARD s) (h4/pred__set/CARD t))
% Assm: h4/real/real__div: !y x. h4/real/_2F x y = h4/realax/real__mul x (h4/realax/inv y)
% 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__LE__INV: !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) (h4/realax/inv x)
% Assm: h4/real/REAL__ADD: !n m. h4/realax/real__add (h4/real/real__of__num m) (h4/real/real__of__num n) = h4/real/real__of__num (h4/arithmetic/_2B m n)
% Assm: h4/real/REAL__DIV__REFL: !x. ~(x = h4/real/real__of__num h4/num/0) ==> h4/real/_2F x x = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/real/REAL__DIV__LZERO: !x. h4/real/_2F (h4/real/real__of__num h4/num/0) x = h4/real/real__of__num h4/num/0
% 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/eq__ints_c0: !n m. h4/real/real__of__num n = h4/real/real__of__num m <=> n = m
% Assm: h4/real/le__int_c0: !n m. h4/real/real__lte (h4/real/real__of__num n) (h4/real/real__of__num m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/util__prob/IN__FUNSET: !f Q P. h4/bool/IN f (h4/util__prob/FUNSET P Q) <=> (!x. h4/bool/IN x P ==> h4/bool/IN (f x) Q)
% Assm: h4/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/subset__class__def: !sts sp. h4/measure/subset__class sp sts <=> (!x. h4/bool/IN x sts ==> h4/pred__set/SUBSET x sp)
% Assm: h4/measure/algebra__def: !a. h4/measure/algebra a <=> h4/measure/subset__class (h4/measure/space a) (h4/measure/subsets a) /\ h4/bool/IN h4/pred__set/EMPTY (h4/measure/subsets a) /\ (!s. h4/bool/IN s (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/DIFF (h4/measure/space a) s) (h4/measure/subsets a)) /\ (!s t. h4/bool/IN s (h4/measure/subsets a) /\ h4/bool/IN t (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/UNION s t) (h4/measure/subsets a))
% Assm: h4/measure/sigma__algebra__def: !a. h4/measure/sigma__algebra a <=> h4/measure/algebra a /\ (!c. h4/util__prob/countable c /\ h4/pred__set/SUBSET c (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/BIGUNION c) (h4/measure/subsets a))
% 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/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/IN__MEASURABLE: !f b a. h4/bool/IN f (h4/measure/measurable a b) <=> h4/measure/sigma__algebra a /\ h4/measure/sigma__algebra b /\ h4/bool/IN f (h4/util__prob/FUNSET (h4/measure/space a) (h4/measure/space b)) /\ (!s. h4/bool/IN s (h4/measure/subsets b) ==> h4/bool/IN (h4/pred__set/INTER (h4/util__prob/PREIMAGE f s) (h4/measure/space a)) (h4/measure/subsets a))
% Assm: h4/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/probability/prob__space__def: !p. h4/probability/prob__space p <=> h4/measure/measure__space p /\ h4/measure/measure p (h4/probability/p__space p) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/probability/random__variable__def: !s p X. h4/probability/random__variable X p s <=> h4/probability/prob__space p /\ h4/bool/IN X (h4/measure/measurable (h4/pair/_2C (h4/probability/p__space p) (h4/probability/events p)) s)
% Assm: h4/probability/uniform__distribution__def: !s p X. h4/probability/uniform__distribution p X s = (\a. h4/real/_2F (h4/real/real__of__num (h4/pred__set/CARD a)) (h4/real/real__of__num (h4/pred__set/CARD (h4/measure/space s))))
% Assm: h4/probability/PSPACE: !c b a. h4/probability/p__space (h4/pair/_2C a (h4/pair/_2C b c)) = a
% Goal: !s p X. h4/pred__set/FINITE (h4/probability/p__space p) /\ h4/pred__set/FINITE (h4/measure/space s) /\ h4/probability/random__variable X p s ==> h4/probability/prob__space (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) (h4/probability/uniform__distribution p X s)))
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 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_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> 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_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% 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_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% 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_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_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_numerals_numeralu_u_distribu_c18]: !n. h4/num/0 = h4/arithmetic/NUMERAL n <=> n = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% 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_CHOICEu_u_DEF]: !s. ~(s = h4/pred__set/EMPTY) ==> h4/bool/IN (h4/pred__set/CHOICE s) 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_CARDu_u_EMPTY]: h4/pred__set/CARD h4/pred__set/EMPTY = h4/num/0
% Assm [h4s_predu_u_sets_CARDu_u_EQu_u_0]: !s. h4/pred__set/FINITE s ==> (h4/pred__set/CARD s = h4/num/0 <=> s = h4/pred__set/EMPTY)
% Assm [h4s_predu_u_sets_CARDu_u_UNION]: !s. h4/pred__set/FINITE s ==> (!t. h4/pred__set/FINITE t ==> h4/arithmetic/_2B (h4/pred__set/CARD (h4/pred__set/UNION s t)) (h4/pred__set/CARD (h4/pred__set/INTER s t)) = h4/arithmetic/_2B (h4/pred__set/CARD s) (h4/pred__set/CARD t))
% Assm [h4s_reals_realu_u_div]: !y x. h4/real/_2F x y = h4/realax/real__mul x (h4/realax/inv y)
% 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_LEu_u_INV]: !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) (h4/realax/inv x)
% Assm [h4s_reals_REALu_u_ADD]: !n m. h4/realax/real__add (h4/real/real__of__num m) (h4/real/real__of__num n) = h4/real/real__of__num (h4/arithmetic/_2B m n)
% Assm [h4s_reals_REALu_u_DIVu_u_REFL]: !x. ~(x = h4/real/real__of__num h4/num/0) ==> h4/real/_2F x x = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_reals_REALu_u_DIVu_u_LZERO]: !x. h4/real/_2F (h4/real/real__of__num h4/num/0) x = h4/real/real__of__num h4/num/0
% 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_reals_equ_u_intsu_c0]: !n m. h4/real/real__of__num n = h4/real/real__of__num m <=> n = m
% Assm [h4s_reals_leu_u_intu_c0]: !n m. h4/real/real__lte (h4/real/real__of__num n) (h4/real/real__of__num m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_utilu_u_probs_INu_u_FUNSET]: !f Q P. h4/bool/IN f (h4/util__prob/FUNSET P Q) <=> (!x. h4/bool/IN x P ==> h4/bool/IN (happ f x) Q)
% Assm [h4s_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_subsetu_u_classu_u_def]: !sts sp. h4/measure/subset__class sp sts <=> (!x. h4/bool/IN x sts ==> h4/pred__set/SUBSET x sp)
% Assm [h4s_measures_algebrau_u_def]: !a. h4/measure/algebra a <=> h4/measure/subset__class (h4/measure/space a) (h4/measure/subsets a) /\ h4/bool/IN h4/pred__set/EMPTY (h4/measure/subsets a) /\ (!s. h4/bool/IN s (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/DIFF (h4/measure/space a) s) (h4/measure/subsets a)) /\ (!s t. h4/bool/IN s (h4/measure/subsets a) /\ h4/bool/IN t (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/UNION s t) (h4/measure/subsets a))
% Assm [h4s_measures_sigmau_u_algebrau_u_def]: !a. h4/measure/sigma__algebra a <=> h4/measure/algebra a /\ (!c. h4/util__prob/countable c /\ h4/pred__set/SUBSET c (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/BIGUNION c) (h4/measure/subsets a))
% 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_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_INu_u_MEASURABLE]: !f b a. h4/bool/IN f (h4/measure/measurable a b) <=> h4/measure/sigma__algebra a /\ h4/measure/sigma__algebra b /\ h4/bool/IN f (h4/util__prob/FUNSET (h4/measure/space a) (h4/measure/space b)) /\ (!s. h4/bool/IN s (h4/measure/subsets b) ==> h4/bool/IN (h4/pred__set/INTER (h4/util__prob/PREIMAGE f s) (h4/measure/space a)) (h4/measure/subsets a))
% Assm [h4s_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_probabilitys_probu_u_spaceu_u_def]: !p. h4/probability/prob__space p <=> h4/measure/measure__space p /\ happ (h4/measure/measure p) (h4/probability/p__space p) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_probabilitys_randomu_u_variableu_u_def]: !s p X. h4/probability/random__variable X p s <=> h4/probability/prob__space p /\ h4/bool/IN X (h4/measure/measurable (h4/pair/_2C (h4/probability/p__space p) (h4/probability/events p)) s)
% Assm [h4s_probabilitys_uniformu_u_distributionu_u_def]: !s p X x. happ (h4/probability/uniform__distribution p X s) x = h4/real/_2F (h4/real/real__of__num (h4/pred__set/CARD x)) (h4/real/real__of__num (h4/pred__set/CARD (h4/measure/space s)))
% Assm [h4s_probabilitys_PSPACE]: !c b a. h4/probability/p__space (h4/pair/_2C a (h4/pair/_2C b c)) = a
% Goal: !s p X. h4/pred__set/FINITE (h4/probability/p__space p) /\ h4/pred__set/FINITE (h4/measure/space s) /\ h4/probability/random__variable X p s ==> h4/probability/prob__space (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) (h4/probability/uniform__distribution p X s)))
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_Q223756,TV_Q223752]: ![V_f, V_g]: (![V_x]: s(TV_Q223752,happ(s(t_fun(TV_Q223756,TV_Q223752),V_f),s(TV_Q223756,V_x))) = s(TV_Q223752,happ(s(t_fun(TV_Q223756,TV_Q223752),V_g),s(TV_Q223756,V_x))) => s(t_fun(TV_Q223756,TV_Q223752),V_f) = s(t_fun(TV_Q223756,TV_Q223752),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_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_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_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_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_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(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_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
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_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_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_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_NOTu_u_LEQ, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_c18, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_nums_0) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_equ_c1, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero) <=> p(s(t_bool,f)))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
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_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_CHOICEu_u_DEF, axiom, ![TV_u_27a]: ![V_s]: (~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))),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_CARDu_u_EMPTY, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_predu_u_sets_CARDu_u_EQu_u_0, 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)))) => (s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_nums_num,h4s_nums_0) <=> s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))).
fof(ah4s_predu_u_sets_CARDu_u_UNION, 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_finite(s(t_fun(TV_u_27a,t_bool),V_t)))) => s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_predu_u_sets_card(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_nums_num,h4s_predu_u_sets_card(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_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_t)))))))).
fof(ah4s_reals_realu_u_div, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_inv(s(t_h4s_realaxs_real,V_y)))))).
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_LEu_u_INV, axiom, ![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,h4s_realaxs_inv(s(t_h4s_realaxs_real,V_x)))))))).
fof(ah4s_reals_REALu_u_ADD, axiom, ![V_n, V_m]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_reals_REALu_u_DIVu_u_REFL, axiom, ![V_x]: (~ (s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x),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_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))).
fof(ah4s_reals_REALu_u_DIVu_u_LZERO, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_u_2f(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_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_reals_equ_u_intsu_c0, axiom, ![V_n, V_m]: (s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,V_m))).
fof(ah4s_reals_leu_u_intu_c0, axiom, ![V_n, V_m]: s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_utilu_u_probs_INu_u_FUNSET, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_Q, V_P]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),h4s_utilu_u_probs_funset(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_Q))))))).
fof(ah4s_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_subsetu_u_classu_u_def, axiom, ![TV_u_27a]: ![V_sts, V_sp]: (p(s(t_bool,h4s_measures_subsetu_u_class(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts)))) => p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(TV_u_27a,t_bool),V_sp))))))).
fof(ah4s_measures_algebrau_u_def, axiom, ![TV_u_27a]: ![V_a]: (p(s(t_bool,h4s_measures_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))) <=> (p(s(t_bool,h4s_measures_subsetu_u_class(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))) & (![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))))))) & ![V_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_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))))))))).
fof(ah4s_measures_sigmau_u_algebrau_u_def, axiom, ![TV_u_27a]: ![V_a]: (p(s(t_bool,h4s_measures_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))) <=> (p(s(t_bool,h4s_measures_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))) & ![V_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_a))))))) => 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_a)))))))))).
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_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_INu_u_MEASURABLE, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_b, V_a]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_b)))))) <=> (p(s(t_bool,h4s_measures_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))) & (p(s(t_bool,h4s_measures_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_b)))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),h4s_utilu_u_probs_funset(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))),s(t_fun(TV_u_27b,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_b)))))))) & ![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27b,t_bool),V_s),s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_b)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27b,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))))))))).
fof(ah4s_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_probabilitys_probu_u_spaceu_u_def, axiom, ![TV_u_27a]: ![V_p]: (p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) <=> (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) & s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),h4s_probabilitys_pu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))).
fof(ah4s_probabilitys_randomu_u_variableu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_s, V_p, V_X]: (p(s(t_bool,h4s_probabilitys_randomu_u_variable(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s)))) <=> (p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_probabilitys_pu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_probabilitys_events(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s))))))))).
fof(ah4s_probabilitys_uniformu_u_distributionu_u_def, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a,TV_u_27d]: ![V_s, V_p, V_X, V_x]: s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_probabilitys_uniformu_u_distribution(s(TV_u_27b,V_p),s(TV_u_27c,V_X),s(t_h4s_pairs_prod(t_fun(TV_u_27d,t_bool),t_fun(t_fun(TV_u_27d,t_bool),t_bool)),V_s))),s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_x))))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27d,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27d,t_bool),t_fun(t_fun(TV_u_27d,t_bool),t_bool)),V_s)))))))))).
fof(ah4s_probabilitys_PSPACE, axiom, ![TV_u_27a]: ![V_c, V_b, V_a]: s(t_fun(TV_u_27a,t_bool),h4s_probabilitys_pu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_a),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_b),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_c))))))) = s(t_fun(TV_u_27a,t_bool),V_a)).
fof(ch4s_probabilitys_uniformu_u_distributionu_u_probu_u_space, conjecture, ![TV_u_27a,TV_u_27b]: ![V_s, V_p, V_X]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_probabilitys_pu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s)))))) & p(s(t_bool,h4s_probabilitys_randomu_u_variable(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s)))))) => p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s))),s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),h4s_probabilitys_uniformu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s)))))))))))).
