%   ORIGINAL: h4/probability/PROB__POSITIVE
% 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/TRUTH: T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/probability/POSITIVE__PROB: !p. h4/measure/positive p <=> h4/probability/prob p h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0 /\ (!s. h4/bool/IN s (h4/probability/events p) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/probability/prob p s))
% Assm: h4/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/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/probability/events__def: h4/probability/events = h4/measure/measurable__sets
% Assm: h4/probability/prob__def: h4/probability/prob = h4/measure/measure
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/probability/PROB__SPACE: !p. h4/probability/prob__space p <=> h4/measure/sigma__algebra (h4/pair/_2C (h4/probability/p__space p) (h4/probability/events p)) /\ h4/measure/positive p /\ h4/measure/countably__additive p /\ h4/probability/prob p (h4/probability/p__space p) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/probability/INCREASING__PROB: !p. h4/measure/increasing p <=> (!s t. h4/bool/IN s (h4/probability/events p) /\ h4/bool/IN t (h4/probability/events p) /\ h4/pred__set/SUBSET s t ==> h4/real/real__lte (h4/probability/prob p s) (h4/probability/prob p t))
% Assm: h4/probability/possibly__def: !p e. h4/probability/possibly p e <=> h4/bool/IN e (h4/probability/events p) /\ ~(h4/probability/prob p e = h4/real/real__of__num h4/num/0)
% Assm: h4/probability/probably__def: !p e. h4/probability/probably p e <=> h4/bool/IN e (h4/probability/events p) /\ h4/probability/prob p e = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/probability/indep__def: !p b a. h4/probability/indep p a b <=> h4/bool/IN a (h4/probability/events p) /\ h4/bool/IN b (h4/probability/events p) /\ h4/probability/prob p (h4/pred__set/INTER a b) = h4/realax/real__mul (h4/probability/prob p a) (h4/probability/prob p b)
% Assm: h4/probability/ADDITIVE__PROB: !p. h4/measure/additive p <=> (!s t. h4/bool/IN s (h4/probability/events p) /\ h4/bool/IN t (h4/probability/events p) /\ h4/pred__set/DISJOINT s t ==> h4/probability/prob p (h4/pred__set/UNION s t) = h4/realax/real__add (h4/probability/prob p s) (h4/probability/prob p t))
% 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/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/probability/EVENTS__SIGMA__ALGEBRA: !p. h4/probability/prob__space p ==> h4/measure/sigma__algebra (h4/pair/_2C (h4/probability/p__space p) (h4/probability/events p))
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/probability/EVENTS__ALGEBRA: !p. h4/probability/prob__space p ==> h4/measure/algebra (h4/pair/_2C (h4/probability/p__space p) (h4/probability/events p))
% 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/real__random__variable__def: !p X. h4/probability/real__random__variable X p <=> h4/probability/prob__space p /\ (!x. h4/bool/IN x (h4/probability/p__space p) ==> ~(X x = h4/extreal/NegInf) /\ ~(X x = h4/extreal/PosInf)) /\ h4/bool/IN X (h4/measure/measurable (h4/pair/_2C (h4/probability/p__space p) (h4/probability/events p)) h4/measure/Borel)
% Assm: h4/probability/PROB__EMPTY: !p. h4/probability/prob__space p ==> h4/probability/prob p h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0
% Assm: h4/probability/PROB__UNIV: !p. h4/probability/prob__space p ==> h4/probability/prob p (h4/probability/p__space p) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/probability/COUNTABLY__ADDITIVE__PROB: !p. h4/measure/countably__additive p <=> (!f. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (h4/probability/events p)) /\ (!m n. ~(m = n) ==> h4/pred__set/DISJOINT (f m) (f n)) /\ h4/bool/IN (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV)) (h4/probability/events p) ==> h4/seq/sums (h4/combin/o (h4/probability/prob p) f) (h4/probability/prob p (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV))))
% 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/PROB__SPACE__POSITIVE: !p. h4/probability/prob__space p ==> h4/measure/positive p
% Assm: h4/measure/increasing__def: !m. h4/measure/increasing m <=> (!s t. h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/bool/IN t (h4/measure/measurable__sets m) /\ h4/pred__set/SUBSET s t ==> h4/real/real__lte (h4/measure/measure m s) (h4/measure/measure m t))
% 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/probability/PROB__SPACE__COUNTABLY__ADDITIVE: !p. h4/probability/prob__space p ==> h4/measure/countably__additive p
% Assm: h4/probability/PROB__SPACE__ADDITIVE: !p. h4/probability/prob__space p ==> h4/measure/additive p
% Assm: h4/probability/PROB__SPACE__INCREASING: !p. h4/probability/prob__space p ==> h4/measure/increasing p
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__disj: !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/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/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/measure/countably__additive__def: !m. h4/measure/countably__additive m <=> (!f. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (h4/measure/measurable__sets m)) /\ (!m0 n. ~(m0 = n) ==> h4/pred__set/DISJOINT (f m0) (f n)) /\ h4/bool/IN (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV)) (h4/measure/measurable__sets m) ==> h4/seq/sums (h4/combin/o (h4/measure/measure m) f) (h4/measure/measure m (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV))))
% Assm: h4/probability/p__space__def: h4/probability/p__space = h4/measure/m__space
% Assm: h4/probability/distribution__def: !p X. h4/probability/distribution p X = (\s. h4/probability/prob p (h4/pred__set/INTER (h4/util__prob/PREIMAGE X s) (h4/probability/p__space p)))
% Assm: h4/probability/joint__distribution3__def: !p Z Y X. h4/probability/joint__distribution3 p X Y Z = (\a. h4/probability/prob p (h4/pred__set/INTER (h4/util__prob/PREIMAGE (\x. h4/pair/_2C (X x) (h4/pair/_2C (Y x) (Z x))) a) (h4/probability/p__space p)))
% Assm: h4/probability/joint__distribution__def: !p Y X. h4/probability/joint__distribution p X Y = (\a. h4/probability/prob p (h4/pred__set/INTER (h4/util__prob/PREIMAGE (\x. h4/pair/_2C (X x) (Y x)) a) (h4/probability/p__space p)))
% Assm: h4/measure/measure__space__def: !m. h4/measure/measure__space m <=> h4/measure/sigma__algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/countably__additive m
% Assm: h4/measure/SIGMA__ALGEBRA__ALGEBRA: !a. h4/measure/sigma__algebra a ==> h4/measure/algebra a
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/measure/COUNTABLY__ADDITIVE__ADDITIVE: !m. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/countably__additive m ==> h4/measure/additive m
% Assm: h4/measure/ADDITIVE__INCREASING: !m. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/additive m ==> h4/measure/increasing m
% Assm: h4/measure/INF__MEASURE__POS0: !x m g. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/bool/IN x (h4/pred__set/GSPEC (\r. h4/pair/_2C r (?f. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (h4/measure/measurable__sets m)) /\ (!m0 n. ~(m0 = n) ==> h4/pred__set/DISJOINT (f m0) (f n)) /\ h4/pred__set/SUBSET g (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV)) /\ h4/seq/sums (h4/combin/o (h4/measure/measure m) f) r))) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) x
% Assm: h4/lebesgue/negative__set__def: !m A. h4/lebesgue/negative__set m A <=> h4/bool/IN A (h4/measure/measurable__sets m) /\ (!s. h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/pred__set/SUBSET s A ==> h4/real/real__lte (h4/measure/measure m s) (h4/real/real__of__num h4/num/0))
% Assm: h4/lebesgue/pos__fn__integral__cmul__indicator: !s m c. h4/measure/measure__space m /\ h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/real/real__lte (h4/real/real__of__num h4/num/0) c ==> h4/lebesgue/pos__fn__integral m (\x. h4/extreal/extreal__mul (h4/extreal/Normal c) (h4/measure/indicator__fn s x)) = h4/extreal/Normal (h4/realax/real__mul c (h4/measure/measure m s))
% Assm: h4/measure/INF__MEASURE__CLOSE: !s m e. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/realax/real__lt (h4/real/real__of__num h4/num/0) e /\ h4/pred__set/SUBSET s (h4/measure/m__space m) ==> (?f l. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (h4/measure/measurable__sets m)) /\ h4/pred__set/SUBSET s (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV)) /\ (!m0 n. ~(m0 = n) ==> h4/pred__set/DISJOINT (f m0) (f n)) /\ h4/seq/sums (h4/combin/o (h4/measure/measure m) f) l /\ h4/real/real__lte l (h4/realax/real__add (h4/measure/inf__measure m s) e))
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/util__prob/SUMS__EQ: !x f. h4/seq/sums f x <=> h4/seq/summable f /\ h4/seq/suminf f = x
% Assm: h4/util__prob/SER__POS: !f. h4/seq/summable f /\ (!n. h4/real/real__lte (h4/real/real__of__num h4/num/0) (f n)) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/seq/suminf f)
% Assm: h4/util__prob/IN__FUNSET: !f Q P. h4/bool/IN f (h4/util__prob/FUNSET P Q) <=> (!x. h4/bool/IN x P ==> h4/bool/IN (f x) Q)
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/lebesgue/pos__simple__fn__integral__indicator: !m A. h4/measure/measure__space m /\ h4/bool/IN A (h4/measure/measurable__sets m) ==> (?s a x. h4/measure/pos__simple__fn m (h4/measure/indicator__fn A) s a x /\ h4/lebesgue/pos__simple__fn__integral m s a x = h4/extreal/Normal (h4/measure/measure m A))
% Assm: h4/lebesgue/pos__fn__integral__pos__simple__fn: !x s m f a. h4/measure/measure__space m /\ h4/measure/pos__simple__fn m f s a x ==> h4/lebesgue/pos__fn__integral m f = h4/lebesgue/pos__simple__fn__integral m s a x
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/lebesgue/pos__fn__integral__cmul: !m f c. h4/measure/measure__space m /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) /\ h4/real/real__lte (h4/real/real__of__num h4/num/0) c ==> h4/lebesgue/pos__fn__integral m (\x. h4/extreal/extreal__mul (h4/extreal/Normal c) (f x)) = h4/extreal/extreal__mul (h4/extreal/Normal c) (h4/lebesgue/pos__fn__integral m f)
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/measure/indicator__fn__def: !s. h4/measure/indicator__fn s = (\x. h4/bool/COND (h4/bool/IN x s) (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/extreal/extreal__of__num h4/num/0))
% Assm: h4/extreal/le__01: h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm: h4/extreal/le__refl: !x. h4/extreal/extreal__le x x
% Assm: h4/extreal/extreal__mul__def_c8: !y x. h4/extreal/extreal__mul (h4/extreal/Normal x) (h4/extreal/Normal y) = h4/extreal/Normal (h4/realax/real__mul x y)
% Assm: h4/extreal/extreal__mul__def_c3: h4/extreal/extreal__mul h4/extreal/PosInf h4/extreal/PosInf = h4/extreal/PosInf
% Assm: h4/extreal/extreal__mul__def_c2: h4/extreal/extreal__mul h4/extreal/PosInf h4/extreal/NegInf = h4/extreal/NegInf
% Assm: h4/extreal/extreal__mul__def_c4: !x. h4/extreal/extreal__mul (h4/extreal/Normal x) h4/extreal/NegInf = h4/bool/COND (x = h4/real/real__of__num h4/num/0) (h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) x) h4/extreal/NegInf h4/extreal/PosInf)
% Assm: h4/extreal/extreal__mul__def_c1: h4/extreal/extreal__mul h4/extreal/NegInf h4/extreal/PosInf = h4/extreal/NegInf
% Assm: h4/extreal/extreal__mul__def_c7: !y. h4/extreal/extreal__mul h4/extreal/PosInf (h4/extreal/Normal y) = h4/bool/COND (y = h4/real/real__of__num h4/num/0) (h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) y) h4/extreal/PosInf h4/extreal/NegInf)
% Assm: h4/extreal/extreal__mul__def_c6: !x. h4/extreal/extreal__mul (h4/extreal/Normal x) h4/extreal/PosInf = h4/bool/COND (x = h4/real/real__of__num h4/num/0) (h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) x) h4/extreal/PosInf h4/extreal/NegInf)
% Assm: h4/extreal/extreal__mul__def_c5: !y. h4/extreal/extreal__mul h4/extreal/NegInf (h4/extreal/Normal y) = h4/bool/COND (y = h4/real/real__of__num h4/num/0) (h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) y) h4/extreal/NegInf h4/extreal/PosInf)
% Assm: h4/extreal/extreal__mul__def_c0: h4/extreal/extreal__mul h4/extreal/NegInf h4/extreal/NegInf = h4/extreal/PosInf
% Assm: h4/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/sat/dc__cond: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% 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/COND__RAND: !y x f b. f (h4/bool/COND b x y) = h4/bool/COND b (f x) (f y)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/measure/INF__MEASURE__NONEMPTY: !s m g. h4/measure/algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/pred__set/SUBSET g s ==> h4/bool/IN (h4/measure/measure m s) (h4/pred__set/GSPEC (\r. h4/pair/_2C r (?f. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (h4/measure/measurable__sets m)) /\ (!m0 n. ~(m0 = n) ==> h4/pred__set/DISJOINT (f m0) (f n)) /\ h4/pred__set/SUBSET g (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV)) /\ h4/seq/sums (h4/combin/o (h4/measure/measure m) f) r)))
% Assm: h4/measure/ALGEBRA__SPACE: !a. h4/measure/algebra a ==> h4/bool/IN (h4/measure/space a) (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/m__space__def: !sts sp mu. h4/measure/m__space (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sp
% Assm: h4/measure/subsets__def: !y x. h4/measure/subsets (h4/pair/_2C x y) = y
% Assm: h4/measure/space__def: !y x. h4/measure/space (h4/pair/_2C x y) = x
% Goal: !s p. h4/probability/prob__space p /\ h4/bool/IN s (h4/probability/events p) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/probability/prob p s)
%   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_TRUTH]: T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_probabilitys_POSITIVEu_u_PROB]: !p. h4/measure/positive p <=> happ (happ h4/probability/prob p) h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0 /\ (!s. h4/bool/IN s (happ h4/probability/events p) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ (happ h4/probability/prob p) s))
% Assm [h4s_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_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_probabilitys_eventsu_u_def]: h4/probability/events = h4/measure/measurable__sets
% Assm [h4s_probabilitys_probu_u_def]: h4/probability/prob = h4/measure/measure
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_probabilitys_PROBu_u_SPACE]: !p. h4/probability/prob__space p <=> h4/measure/sigma__algebra (h4/pair/_2C (happ h4/probability/p__space p) (happ h4/probability/events p)) /\ h4/measure/positive p /\ h4/measure/countably__additive p /\ happ (happ h4/probability/prob p) (happ h4/probability/p__space p) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_probabilitys_INCREASINGu_u_PROB]: !p. h4/measure/increasing p <=> (!s t. h4/bool/IN s (happ h4/probability/events p) /\ h4/bool/IN t (happ h4/probability/events p) /\ h4/pred__set/SUBSET s t ==> h4/real/real__lte (happ (happ h4/probability/prob p) s) (happ (happ h4/probability/prob p) t))
% Assm [h4s_probabilitys_possiblyu_u_def]: !p e. h4/probability/possibly p e <=> h4/bool/IN e (happ h4/probability/events p) /\ ~(happ (happ h4/probability/prob p) e = h4/real/real__of__num h4/num/0)
% Assm [h4s_probabilitys_probablyu_u_def]: !p e. h4/probability/probably p e <=> h4/bool/IN e (happ h4/probability/events p) /\ happ (happ h4/probability/prob p) e = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_probabilitys_indepu_u_def]: !p b a. h4/probability/indep p a b <=> h4/bool/IN a (happ h4/probability/events p) /\ h4/bool/IN b (happ h4/probability/events p) /\ happ (happ h4/probability/prob p) (h4/pred__set/INTER a b) = h4/realax/real__mul (happ (happ h4/probability/prob p) a) (happ (happ h4/probability/prob p) b)
% Assm [h4s_probabilitys_ADDITIVEu_u_PROB]: !p. h4/measure/additive p <=> (!s t. h4/bool/IN s (happ h4/probability/events p) /\ h4/bool/IN t (happ h4/probability/events p) /\ h4/pred__set/DISJOINT s t ==> happ (happ h4/probability/prob p) (h4/pred__set/UNION s t) = h4/realax/real__add (happ (happ h4/probability/prob p) s) (happ (happ h4/probability/prob p) t))
% Assm [h4s_measures_positiveu_u_def]: !m. h4/measure/positive m <=> happ (happ h4/measure/measure m) h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0 /\ (!s. h4/bool/IN s (happ h4/measure/measurable__sets m) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ (happ h4/measure/measure m) s))
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_probabilitys_EVENTSu_u_SIGMAu_u_ALGEBRA]: !p. h4/probability/prob__space p ==> h4/measure/sigma__algebra (h4/pair/_2C (happ h4/probability/p__space p) (happ h4/probability/events p))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_probabilitys_EVENTSu_u_ALGEBRA]: !p. h4/probability/prob__space p ==> h4/measure/algebra (h4/pair/_2C (happ h4/probability/p__space p) (happ h4/probability/events p))
% Assm [h4s_probabilitys_randomu_u_variableu_u_def]: !s p X. h4/probability/random__variable X p s <=> h4/probability/prob__space p /\ h4/bool/IN X (h4/measure/measurable (h4/pair/_2C (happ h4/probability/p__space p) (happ h4/probability/events p)) s)
% Assm [h4s_probabilitys_realu_u_randomu_u_variableu_u_def]: !p X. h4/probability/real__random__variable X p <=> h4/probability/prob__space p /\ (!x. h4/bool/IN x (happ h4/probability/p__space p) ==> ~(happ X x = h4/extreal/NegInf) /\ ~(happ X x = h4/extreal/PosInf)) /\ h4/bool/IN X (h4/measure/measurable (h4/pair/_2C (happ h4/probability/p__space p) (happ h4/probability/events p)) h4/measure/Borel)
% Assm [h4s_probabilitys_PROBu_u_EMPTY]: !p. h4/probability/prob__space p ==> happ (happ h4/probability/prob p) h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0
% Assm [h4s_probabilitys_PROBu_u_UNIV]: !p. h4/probability/prob__space p ==> happ (happ h4/probability/prob p) (happ h4/probability/p__space p) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_probabilitys_COUNTABLYu_u_ADDITIVEu_u_PROB]: !p. h4/measure/countably__additive p <=> (!f. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (happ h4/probability/events p)) /\ (!m n. ~(m = n) ==> h4/pred__set/DISJOINT (happ f m) (happ f n)) /\ h4/bool/IN (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV)) (happ h4/probability/events p) ==> h4/seq/sums (h4/combin/o (happ h4/probability/prob p) f) (happ (happ h4/probability/prob p) (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV))))
% Assm [h4s_probabilitys_probu_u_spaceu_u_def]: !p. h4/probability/prob__space p <=> h4/measure/measure__space p /\ happ (happ h4/measure/measure p) (happ h4/probability/p__space p) = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_probabilitys_PROBu_u_SPACEu_u_POSITIVE]: !p. h4/probability/prob__space p ==> h4/measure/positive p
% Assm [h4s_measures_increasingu_u_def]: !m. h4/measure/increasing m <=> (!s t. h4/bool/IN s (happ h4/measure/measurable__sets m) /\ h4/bool/IN t (happ h4/measure/measurable__sets m) /\ h4/pred__set/SUBSET s t ==> h4/real/real__lte (happ (happ h4/measure/measure m) s) (happ (happ h4/measure/measure m) t))
% Assm [h4s_measures_additiveu_u_def]: !m. h4/measure/additive m <=> (!s t. h4/bool/IN s (happ h4/measure/measurable__sets m) /\ h4/bool/IN t (happ h4/measure/measurable__sets m) /\ h4/pred__set/DISJOINT s t ==> happ (happ h4/measure/measure m) (h4/pred__set/UNION s t) = h4/realax/real__add (happ (happ h4/measure/measure m) s) (happ (happ h4/measure/measure m) t))
% Assm [h4s_probabilitys_PROBu_u_SPACEu_u_COUNTABLYu_u_ADDITIVE]: !p. h4/probability/prob__space p ==> h4/measure/countably__additive p
% Assm [h4s_probabilitys_PROBu_u_SPACEu_u_ADDITIVE]: !p. h4/probability/prob__space p ==> h4/measure/additive p
% Assm [h4s_probabilitys_PROBu_u_SPACEu_u_INCREASING]: !p. h4/probability/prob__space p ==> h4/measure/increasing p
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_disj]: !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_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_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_measures_countablyu_u_additiveu_u_def]: !m. h4/measure/countably__additive m <=> (!f. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (happ h4/measure/measurable__sets m)) /\ (!m0 n. ~(m0 = n) ==> h4/pred__set/DISJOINT (happ f m0) (happ f n)) /\ h4/bool/IN (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV)) (happ h4/measure/measurable__sets m) ==> h4/seq/sums (h4/combin/o (happ h4/measure/measure m) f) (happ (happ h4/measure/measure m) (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV))))
% Assm [h4s_probabilitys_pu_u_spaceu_u_def]: h4/probability/p__space = h4/measure/m__space
% Assm [h4s_probabilitys_distributionu_u_def]: !p X x. h4/probability/distribution p X x = happ (happ h4/probability/prob p) (h4/pred__set/INTER (h4/util__prob/PREIMAGE X x) (happ h4/probability/p__space p))
% Assm [h4s_probabilitys_jointu_u_distribution3u_u_def]: !_0. (!X Y Z x. happ (happ (happ (happ _0 X) Y) Z) x = h4/pair/_2C (happ X x) (h4/pair/_2C (happ Y x) (happ Z x))) ==> (!p Z Y X x. h4/probability/joint__distribution3 p X Y Z x = happ (happ h4/probability/prob p) (h4/pred__set/INTER (h4/util__prob/PREIMAGE (happ (happ (happ _0 X) Y) Z) x) (happ h4/probability/p__space p)))
% Assm [h4s_probabilitys_jointu_u_distributionu_u_def]: !_0. (!X Y x. happ (happ (happ _0 X) Y) x = h4/pair/_2C (happ X x) (happ Y x)) ==> (!p Y X x. h4/probability/joint__distribution p X Y x = happ (happ h4/probability/prob p) (h4/pred__set/INTER (h4/util__prob/PREIMAGE (happ (happ _0 X) Y) x) (happ h4/probability/p__space p)))
% Assm [h4s_measures_measureu_u_spaceu_u_def]: !m. h4/measure/measure__space m <=> h4/measure/sigma__algebra (h4/pair/_2C (happ h4/measure/m__space m) (happ h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/countably__additive m
% Assm [h4s_measures_SIGMAu_u_ALGEBRAu_u_ALGEBRA]: !a. h4/measure/sigma__algebra a ==> h4/measure/algebra a
% 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_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_measures_COUNTABLYu_u_ADDITIVEu_u_ADDITIVE]: !m. h4/measure/algebra (h4/pair/_2C (happ h4/measure/m__space m) (happ h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/countably__additive m ==> h4/measure/additive m
% Assm [h4s_measures_ADDITIVEu_u_INCREASING]: !m. h4/measure/algebra (h4/pair/_2C (happ h4/measure/m__space m) (happ h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/additive m ==> h4/measure/increasing m
% Assm [h4s_measures_INFu_u_MEASUREu_u_POS0]: !_0. (!g m r. ?v. (v <=> (?f. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (happ h4/measure/measurable__sets m)) /\ (!m0 n. ~(m0 = n) ==> h4/pred__set/DISJOINT (happ f m0) (happ f n)) /\ h4/pred__set/SUBSET g (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV)) /\ h4/seq/sums (h4/combin/o (happ h4/measure/measure m) f) r)) /\ happ (happ (happ _0 g) m) r = h4/pair/_2C r v) ==> (!x m g. h4/measure/algebra (h4/pair/_2C (happ h4/measure/m__space m) (happ h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/bool/IN x (h4/pred__set/GSPEC (happ (happ _0 g) m)) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) x)
% Assm [h4s_lebesgues_negativeu_u_setu_u_def]: !m A. h4/lebesgue/negative__set m A <=> h4/bool/IN A (happ h4/measure/measurable__sets m) /\ (!s. h4/bool/IN s (happ h4/measure/measurable__sets m) /\ h4/pred__set/SUBSET s A ==> h4/real/real__lte (happ (happ h4/measure/measure m) s) (h4/real/real__of__num h4/num/0))
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_cmulu_u_indicator]: !_0. (!c s x. happ (happ (happ _0 c) s) x = h4/extreal/extreal__mul (h4/extreal/Normal c) (happ (h4/measure/indicator__fn s) x)) ==> (!s m c. h4/measure/measure__space m /\ h4/bool/IN s (happ h4/measure/measurable__sets m) /\ h4/real/real__lte (h4/real/real__of__num h4/num/0) c ==> h4/lebesgue/pos__fn__integral m (happ (happ _0 c) s) = h4/extreal/Normal (h4/realax/real__mul c (happ (happ h4/measure/measure m) s)))
% Assm [h4s_measures_INFu_u_MEASUREu_u_CLOSE]: !s m e. h4/measure/algebra (h4/pair/_2C (happ h4/measure/m__space m) (happ h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/realax/real__lt (h4/real/real__of__num h4/num/0) e /\ h4/pred__set/SUBSET s (happ h4/measure/m__space m) ==> (?f l. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (happ h4/measure/measurable__sets m)) /\ h4/pred__set/SUBSET s (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV)) /\ (!m0 n. ~(m0 = n) ==> h4/pred__set/DISJOINT (happ f m0) (happ f n)) /\ h4/seq/sums (h4/combin/o (happ h4/measure/measure m) f) l /\ h4/real/real__lte l (h4/realax/real__add (h4/measure/inf__measure m s) e))
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_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_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_utilu_u_probs_SUMSu_u_EQ]: !x f. h4/seq/sums f x <=> h4/seq/summable f /\ h4/seq/suminf f = x
% Assm [h4s_utilu_u_probs_SERu_u_POS]: !f. h4/seq/summable f /\ (!n. h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ f n)) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/seq/suminf f)
% Assm [h4s_utilu_u_probs_INu_u_FUNSET]: !f Q P. h4/bool/IN f (h4/util__prob/FUNSET P Q) <=> (!x. h4/bool/IN x P ==> h4/bool/IN (happ f x) Q)
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_indicator]: !m A. h4/measure/measure__space m /\ h4/bool/IN A (happ h4/measure/measurable__sets m) ==> (?s a x. h4/measure/pos__simple__fn m (h4/measure/indicator__fn A) s a x /\ h4/lebesgue/pos__simple__fn__integral m s a x = h4/extreal/Normal (happ (happ h4/measure/measure m) A))
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_posu_u_simpleu_u_fn]: !x s m f a. h4/measure/measure__space m /\ h4/measure/pos__simple__fn m f s a x ==> h4/lebesgue/pos__fn__integral m f = h4/lebesgue/pos__simple__fn__integral m s a x
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_cmul]: !_0. (!c f x. happ (happ (happ _0 c) f) x = h4/extreal/extreal__mul (h4/extreal/Normal c) (happ f x)) ==> (!m f c. h4/measure/measure__space m /\ (!x. h4/bool/IN x (happ h4/measure/m__space m) ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) /\ h4/real/real__lte (h4/real/real__of__num h4/num/0) c ==> h4/lebesgue/pos__fn__integral m (happ (happ _0 c) f) = h4/extreal/extreal__mul (h4/extreal/Normal c) (h4/lebesgue/pos__fn__integral m f))
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_measures_indicatoru_u_fnu_u_def]: !s x. happ (h4/measure/indicator__fn s) x = h4/bool/COND (h4/bool/IN x s) (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/extreal/extreal__of__num h4/num/0)
% Assm [h4s_extreals_leu_u_01]: h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_extreals_leu_u_refl]: !x. h4/extreal/extreal__le x x
% Assm [h4s_extreals_extrealu_u_mulu_u_defu_c8]: !y x. h4/extreal/extreal__mul (h4/extreal/Normal x) (h4/extreal/Normal y) = h4/extreal/Normal (h4/realax/real__mul x y)
% Assm [h4s_extreals_extrealu_u_mulu_u_defu_c3]: h4/extreal/extreal__mul h4/extreal/PosInf h4/extreal/PosInf = h4/extreal/PosInf
% Assm [h4s_extreals_extrealu_u_mulu_u_defu_c2]: h4/extreal/extreal__mul h4/extreal/PosInf h4/extreal/NegInf = h4/extreal/NegInf
% Assm [h4s_extreals_extrealu_u_mulu_u_defu_c4]: !x. ?v. (v <=> x = h4/real/real__of__num h4/num/0) /\ h4/extreal/extreal__mul (h4/extreal/Normal x) h4/extreal/NegInf = h4/bool/COND v (h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) x) h4/extreal/NegInf h4/extreal/PosInf)
% Assm [h4s_extreals_extrealu_u_mulu_u_defu_c1]: h4/extreal/extreal__mul h4/extreal/NegInf h4/extreal/PosInf = h4/extreal/NegInf
% Assm [h4s_extreals_extrealu_u_mulu_u_defu_c7]: !y. ?v. (v <=> y = h4/real/real__of__num h4/num/0) /\ h4/extreal/extreal__mul h4/extreal/PosInf (h4/extreal/Normal y) = h4/bool/COND v (h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) y) h4/extreal/PosInf h4/extreal/NegInf)
% Assm [h4s_extreals_extrealu_u_mulu_u_defu_c6]: !x. ?v. (v <=> x = h4/real/real__of__num h4/num/0) /\ h4/extreal/extreal__mul (h4/extreal/Normal x) h4/extreal/PosInf = h4/bool/COND v (h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) x) h4/extreal/PosInf h4/extreal/NegInf)
% Assm [h4s_extreals_extrealu_u_mulu_u_defu_c5]: !y. ?v. (v <=> y = h4/real/real__of__num h4/num/0) /\ h4/extreal/extreal__mul h4/extreal/NegInf (h4/extreal/Normal y) = h4/bool/COND v (h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) y) h4/extreal/NegInf h4/extreal/PosInf)
% Assm [h4s_extreals_extrealu_u_mulu_u_defu_c0]: h4/extreal/extreal__mul h4/extreal/NegInf h4/extreal/NegInf = h4/extreal/PosInf
% Assm [h4s_combins_Iu_u_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_sats_dcu_u_cond]: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% 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_CONDu_u_RAND]: !y x f b. happ f (h4/bool/COND b x y) = h4/bool/COND b (happ f x) (happ f y)
% 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_measures_INFu_u_MEASUREu_u_NONEMPTY]: !_0. (!g m r. ?v. (v <=> (?f. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (happ h4/measure/measurable__sets m)) /\ (!m0 n. ~(m0 = n) ==> h4/pred__set/DISJOINT (happ f m0) (happ f n)) /\ h4/pred__set/SUBSET g (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV)) /\ h4/seq/sums (h4/combin/o (happ h4/measure/measure m) f) r)) /\ happ (happ (happ _0 g) m) r = h4/pair/_2C r v) ==> (!s m g. h4/measure/algebra (h4/pair/_2C (happ h4/measure/m__space m) (happ h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/bool/IN s (happ h4/measure/measurable__sets m) /\ h4/pred__set/SUBSET g s ==> h4/bool/IN (happ (happ h4/measure/measure m) s) (h4/pred__set/GSPEC (happ (happ _0 g) m)))
% Assm [h4s_measures_ALGEBRAu_u_SPACE]: !a. h4/measure/algebra a ==> h4/bool/IN (h4/measure/space a) (h4/measure/subsets a)
% Assm [h4s_measures_measurableu_u_setsu_u_def]: !sts sp mu. happ h4/measure/measurable__sets (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sts
% Assm [h4s_measures_mu_u_spaceu_u_def]: !sts sp mu. happ h4/measure/m__space (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sp
% Assm [h4s_measures_subsetsu_u_def]: !y x. h4/measure/subsets (h4/pair/_2C x y) = y
% Assm [h4s_measures_spaceu_u_def]: !y x. h4/measure/space (h4/pair/_2C x y) = x
% Goal: !s p. h4/probability/prob__space p /\ h4/bool/IN s (happ h4/probability/events p) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ (happ h4/probability/prob p) s)
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_Q1314849,TV_Q1314845]: ![V_f, V_g]: (![V_x]: s(TV_Q1314845,happ(s(t_fun(TV_Q1314849,TV_Q1314845),V_f),s(TV_Q1314849,V_x))) = s(TV_Q1314845,happ(s(t_fun(TV_Q1314849,TV_Q1314845),V_g),s(TV_Q1314849,V_x))) => s(t_fun(TV_Q1314849,TV_Q1314845),V_f) = s(t_fun(TV_Q1314849,TV_Q1314845),V_g))).
fof(ah4s_bools_TRUTH, axiom, 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_probabilitys_POSITIVEu_u_PROB, axiom, ![TV_u_27a]: ![V_p]: (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_p)))) <=> (s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & ![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_s)))))))))).
fof(ah4s_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_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_probabilitys_eventsu_u_def, axiom, ![TV_u_27a]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets)).
fof(ah4s_probabilitys_probu_u_def, axiom, ![TV_u_27a]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure)).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_probabilitys_PROBu_u_SPACE, 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_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),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))))) & (p(s(t_bool,h4s_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_p)))) & (p(s(t_bool,h4s_measures_countablyu_u_additive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) & s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))))).
fof(ah4s_probabilitys_INCREASINGu_u_PROB, axiom, ![TV_u_27a]: ![V_p]: (p(s(t_bool,h4s_measures_increasing(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) <=> ![V_s, V_t]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_t))))))))).
fof(ah4s_probabilitys_possiblyu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e]: (p(s(t_bool,h4s_probabilitys_possibly(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,t_bool),V_e)))) <=> (p(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),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & ~ (s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_e))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_probabilitys_probablyu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e]: (p(s(t_bool,h4s_probabilitys_probably(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,t_bool),V_e)))) <=> (p(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),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_e))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))).
fof(ah4s_probabilitys_indepu_u_def, axiom, ![TV_u_27a]: ![V_p, V_b, V_a]: (p(s(t_bool,h4s_probabilitys_indep(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27a,t_bool),V_a),s(t_fun(TV_u_27a,t_bool),V_b)))) <=> (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_a),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_b),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_a),s(t_fun(TV_u_27a,t_bool),V_b))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_a))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_b))))))))).
fof(ah4s_probabilitys_ADDITIVEu_u_PROB, axiom, ![TV_u_27a]: ![V_p]: (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_p)))) <=> ![V_s, V_t]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) & p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_t)))))))).
fof(ah4s_measures_positiveu_u_def, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) <=> (s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & ![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_s)))))))))).
fof(ah4s_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_probabilitys_EVENTSu_u_SIGMAu_u_ALGEBRA, 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_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),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))))))).
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_probabilitys_EVENTSu_u_ALGEBRA, 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_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),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))))))).
fof(ah4s_probabilitys_randomu_u_variableu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_s, V_p, V_X]: (p(s(t_bool,h4s_probabilitys_randomu_u_variable(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s)))) <=> (p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,TV_u_27b),V_X),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_bool)),V_s))))))))).
fof(ah4s_probabilitys_realu_u_randomu_u_variableu_u_def, axiom, ![TV_u_27a]: ![V_p, V_X]: (p(s(t_bool,h4s_probabilitys_realu_u_randomu_u_variable(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_X),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) <=> (p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) & (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))) => (~ (s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_X),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_neginf)) & ~ (s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_X),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_X),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))))).
fof(ah4s_probabilitys_PROBu_u_EMPTY, axiom, ![TV_u_27a]: ![V_p]: (p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_probabilitys_PROBu_u_UNIV, axiom, ![TV_u_27a]: ![V_p]: (p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))).
fof(ah4s_probabilitys_COUNTABLYu_u_ADDITIVEu_u_PROB, axiom, ![TV_u_27a]: ![V_p]: (p(s(t_bool,h4s_measures_countablyu_u_additive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) <=> ![V_f]: ((p(s(t_bool,h4s_bools_in(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_bool),h4s_utilu_u_probs_funset(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))))) & (![V_m, V_n]: (~ (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)) => p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_h4s_nums_num,V_m))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_h4s_nums_num,V_n))))))) & 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),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))))) => p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))))))))))).
fof(ah4s_probabilitys_probu_u_spaceu_u_def, axiom, ![TV_u_27a]: ![V_p]: (p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) <=> (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) & s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))).
fof(ah4s_probabilitys_PROBu_u_SPACEu_u_POSITIVE, 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_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_p)))))).
fof(ah4s_measures_increasingu_u_def, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_increasing(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) <=> ![V_s, V_t]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_t))))))))).
fof(ah4s_measures_additiveu_u_def, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_additive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) <=> ![V_s, V_t]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_t)))))))).
fof(ah4s_probabilitys_PROBu_u_SPACEu_u_COUNTABLYu_u_ADDITIVE, 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_countablyu_u_additive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))).
fof(ah4s_probabilitys_PROBu_u_SPACEu_u_ADDITIVE, 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_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_p)))))).
fof(ah4s_probabilitys_PROBu_u_SPACEu_u_INCREASING, 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_increasing(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_sats_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_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_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_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_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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_measures_countablyu_u_additiveu_u_def, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_countablyu_u_additive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) <=> ![V_f]: ((p(s(t_bool,h4s_bools_in(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_bool),h4s_utilu_u_probs_funset(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (![V_m0, V_n]: (~ (s(t_h4s_nums_num,V_m0) = s(t_h4s_nums_num,V_n)) => p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_h4s_nums_num,V_m0))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_h4s_nums_num,V_n))))))) & 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),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) => p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))))))))))).
fof(ah4s_probabilitys_pu_u_spaceu_u_def, axiom, ![TV_u_27a]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_probabilitys_pu_u_space) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_measures_mu_u_space)).
fof(ah4s_probabilitys_distributionu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_X, V_x]: s(t_h4s_realaxs_real,h4s_probabilitys_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27b,TV_u_27a),V_X),s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27b,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27b,TV_u_27a),V_X),s(t_fun(TV_u_27a,t_bool),V_x))),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27b,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_p)))))))).
fof(ah4s_probabilitys_jointu_u_distribution3u_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c,TV_u_27d]: ![V_uu_0]: (![V_X, V_Y, V_Z, V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c))))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27a),t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)))))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27a),V_X))),s(t_fun(TV_u_27d,TV_u_27b),V_Y))),s(t_fun(TV_u_27d,TV_u_27c),V_Z))),s(TV_u_27d,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(TV_u_27d,TV_u_27a),V_X),s(TV_u_27d,V_x))),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_Y),s(TV_u_27d,V_x))),s(TV_u_27c,happ(s(t_fun(TV_u_27d,TV_u_27c),V_Z),s(TV_u_27d,V_x))))))) => ![V_p, V_Z, V_Y, V_X, V_x]: s(t_h4s_realaxs_real,h4s_probabilitys_jointu_u_distribution3(s(t_h4s_pairs_prod(t_fun(TV_u_27d,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27d,t_bool),t_bool),t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27d,TV_u_27a),V_X),s(t_fun(TV_u_27d,TV_u_27b),V_Y),s(t_fun(TV_u_27d,TV_u_27c),V_Z),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),V_x))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27d,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27d,t_bool),t_bool),t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27d,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27d,t_bool),t_bool),t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27d,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27d,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c))))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27a),t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27d,TV_u_27c),t_fun(TV_u_27d,t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)))))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27a),V_X))),s(t_fun(TV_u_27d,TV_u_27b),V_Y))),s(t_fun(TV_u_27d,TV_u_27c),V_Z))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27c)),t_bool),V_x))),s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27d,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27d,t_bool),t_bool),t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27d,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27d,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27d,t_bool),t_bool),t_fun(t_fun(TV_u_27d,t_bool),t_h4s_realaxs_real))),V_p))))))))).
fof(ah4s_probabilitys_jointu_u_distributionu_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_X, V_Y, V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_X))),s(t_fun(TV_u_27c,TV_u_27b),V_Y))),s(TV_u_27c,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_X),s(TV_u_27c,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_Y),s(TV_u_27c,V_x))))) => ![V_p, V_Y, V_X, V_x]: s(t_h4s_realaxs_real,h4s_probabilitys_jointu_u_distribution(s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_p),s(t_fun(TV_u_27c,TV_u_27a),V_X),s(t_fun(TV_u_27c,TV_u_27b),V_Y),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_x))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27c,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27c,TV_u_27a),V_X))),s(t_fun(TV_u_27c,TV_u_27b),V_Y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_x))),s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27c,t_bool)),h4s_probabilitys_pu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27c,t_bool),t_bool),t_fun(t_fun(TV_u_27c,t_bool),t_h4s_realaxs_real))),V_p))))))))).
fof(ah4s_measures_measureu_u_spaceu_u_def, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) <=> (p(s(t_bool,h4s_measures_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_measures_countablyu_u_additive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))).
fof(ah4s_measures_SIGMAu_u_ALGEBRAu_u_ALGEBRA, 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)))))).
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_EXISTSu_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,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_CONJu_u_ASSOC, 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_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ?[V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_bools_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_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_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_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_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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_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_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_DISJu_u_ASSOC, 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_C))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_measures_COUNTABLYu_u_ADDITIVEu_u_ADDITIVE, axiom, ![TV_u_27a]: ![V_m]: ((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)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_measures_countablyu_u_additive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => 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)))))).
fof(ah4s_measures_ADDITIVEu_u_INCREASING, axiom, ![TV_u_27a]: ![V_m]: ((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)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_measures_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)))))) => p(s(t_bool,h4s_measures_increasing(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))).
fof(ah4s_measures_INFu_u_MEASUREu_u_POS0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_g, V_m, V_r]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_f]: (p(s(t_bool,h4s_bools_in(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_bool),h4s_utilu_u_probs_funset(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (![V_m0, V_n]: (~ (s(t_h4s_nums_num,V_m0) = s(t_h4s_nums_num,V_n)) => p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_h4s_nums_num,V_m0))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_h4s_nums_num,V_n))))))) & (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_g),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ)))))))) & p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f))),s(t_h4s_realaxs_real,V_r)))))))) & s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_g))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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,V_r))) = s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_r),s(t_bool,V_v)))) => ![V_x, V_m, V_g]: ((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)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_bools_in(s(t_h4s_realaxs_real,V_x),s(t_fun(t_h4s_realaxs_real,t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_g))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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,V_x))))))).
fof(ah4s_lebesgues_negativeu_u_setu_u_def, axiom, ![TV_u_27a]: ![V_m, V_A]: (p(s(t_bool,h4s_lebesgues_negativeu_u_set(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_A)))) <=> (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_A),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & ![V_s]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_A))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_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,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_cmulu_u_indicator, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_c, V_s, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_h4s_realaxs_real,V_c))),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_c))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_indicatoru_u_fn(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_x))))) => ![V_s, V_m, V_c]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & 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_c)))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_h4s_realaxs_real,V_c))),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_c),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_s)))))))))).
fof(ah4s_measures_INFu_u_MEASUREu_u_CLOSE, axiom, ![TV_u_27a]: ![V_s, V_m, V_e]: ((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)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_realaxs_realu_u_lt(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_e)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))))) => ?[V_f, V_l]: (p(s(t_bool,h4s_bools_in(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_bool),h4s_utilu_u_probs_funset(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ)))))))) & (![V_m0, V_n]: (~ (s(t_h4s_nums_num,V_m0) = s(t_h4s_nums_num,V_n)) => p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_h4s_nums_num,V_m0))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_h4s_nums_num,V_n))))))) & (p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f))),s(t_h4s_realaxs_real,V_l)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_measures_infu_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_m),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_realaxs_real,V_e)))))))))))).
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_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_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_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_utilu_u_probs_SUMSu_u_EQ, axiom, ![V_x, V_f]: (p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x)))) <=> (p(s(t_bool,h4s_seqs_summable(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f)))) & s(t_h4s_realaxs_real,h4s_seqs_suminf(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))) = s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_utilu_u_probs_SERu_u_POS, axiom, ![V_f]: ((p(s(t_bool,h4s_seqs_summable(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f)))) & ![V_n]: p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_nums_num,V_n))))))) => 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_seqs_suminf(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f)))))))).
fof(ah4s_utilu_u_probs_INu_u_FUNSET, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_Q, V_P]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),h4s_utilu_u_probs_funset(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_Q))))))).
fof(ah4s_predu_u_sets_INu_u_UNIV, axiom, ![TV_u_27a]: ![V_x]: p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_lebesgues_posu_u_simpleu_u_fnu_u_integralu_u_indicator, axiom, ![TV_u_27a]: ![V_m, V_A]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_A),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))) => ?[V_s, V_a, V_x]: (p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_indicatoru_u_fn(s(t_fun(TV_u_27a,t_bool),V_A))),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x)))) & s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_A)))))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_posu_u_simpleu_u_fn, axiom, ![TV_u_27a]: ![V_x, V_s, V_m, V_f, V_a]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_simpleu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_cmul, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_c, V_f, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_h4s_realaxs_real,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_c))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))))) => ![V_m, V_f, V_c]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))))))) & 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_c)))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_h4s_realaxs_real,V_c))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_c))),s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f)))))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_measures_indicatoru_u_fnu_u_def, axiom, ![TV_u_27a]: ![V_s, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_indicatoru_u_fn(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_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))))))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_extreals_leu_u_01, axiom, p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_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_extreals_leu_u_refl, axiom, ![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_x))))).
fof(ah4s_extreals_extrealu_u_mulu_u_defu_c8, axiom, ![V_y, V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_x))),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_y))))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_extreals_extrealu_u_mulu_u_defu_c3, axiom, s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,h4s_extreals_posinf))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)).
fof(ah4s_extreals_extrealu_u_mulu_u_defu_c2, axiom, s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,h4s_extreals_neginf))) = s(t_h4s_extreals_extreal,h4s_extreals_neginf)).
fof(ah4s_extreals_extrealu_u_mulu_u_defu_c4, axiom, ![V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> 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_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_x))),s(t_h4s_extreals_extreal,h4s_extreals_neginf))) = s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,h4s_realaxs_realu_u_lt(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_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,h4s_extreals_posinf))))))).
fof(ah4s_extreals_extrealu_u_mulu_u_defu_c1, axiom, s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,h4s_extreals_posinf))) = s(t_h4s_extreals_extreal,h4s_extreals_neginf)).
fof(ah4s_extreals_extrealu_u_mulu_u_defu_c7, axiom, ![V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_realaxs_real,V_y) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) & s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_y))))) = s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,h4s_realaxs_realu_u_lt(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))),s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,h4s_extreals_neginf))))))).
fof(ah4s_extreals_extrealu_u_mulu_u_defu_c6, axiom, ![V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> 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_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_x))),s(t_h4s_extreals_extreal,h4s_extreals_posinf))) = s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,h4s_realaxs_realu_u_lt(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_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,h4s_extreals_neginf))))))).
fof(ah4s_extreals_extrealu_u_mulu_u_defu_c5, axiom, ![V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_realaxs_real,V_y) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) & s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_y))))) = s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,h4s_realaxs_realu_u_lt(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))),s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,h4s_extreals_posinf))))))).
fof(ah4s_extreals_extrealu_u_mulu_u_defu_c0, axiom, s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,h4s_extreals_neginf))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)).
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_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_sats_dcu_u_cond, axiom, ![V_s, V_r, V_q, V_p]: (s(t_bool,V_p) = s(t_bool,h4s_bools_cond(s(t_bool,V_q),s(t_bool,V_r),s(t_bool,V_s))) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_s))))) & ((p(s(t_bool,V_p)) | (~ (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_s))))) & ((~ (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_s)) | ~ (p(s(t_bool,V_p))))))))))).
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_CONDu_u_RAND, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_x),s(TV_u_27a,V_y))))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))))).
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_measures_INFu_u_MEASUREu_u_NONEMPTY, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_g, V_m, V_r]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_f]: (p(s(t_bool,h4s_bools_in(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_bool),h4s_utilu_u_probs_funset(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (![V_m0, V_n]: (~ (s(t_h4s_nums_num,V_m0) = s(t_h4s_nums_num,V_n)) => p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_h4s_nums_num,V_m0))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_h4s_nums_num,V_n))))))) & (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_g),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ)))))))) & p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),h4s_combins_o(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_f))),s(t_h4s_realaxs_real,V_r)))))))) & s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_g))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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,V_r))) = s(t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool),h4s_pairs_u_2c(s(t_h4s_realaxs_real,V_r),s(t_bool,V_v)))) => ![V_s, V_m, V_g]: ((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)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_g),s(t_fun(TV_u_27a,t_bool),V_s))))))) => p(s(t_bool,h4s_bools_in(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_measures_measure),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_realaxs_real,t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_h4s_realaxs_real,t_h4s_pairs_prod(t_h4s_realaxs_real,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_g))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))))))).
fof(ah4s_measures_ALGEBRAu_u_SPACE, 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_bools_in(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_measurableu_u_setsu_u_def, axiom, ![TV_u_27a]: ![V_sts, V_sp, V_mu]: s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_measures_measurableu_u_sets),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_mu))))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts)).
fof(ah4s_measures_mu_u_spaceu_u_def, axiom, ![TV_u_27a]: ![V_sts, V_sp, V_mu]: s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_bool)),h4s_measures_mu_u_space),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_mu))))))) = s(t_fun(TV_u_27a,t_bool),V_sp)).
fof(ah4s_measures_subsetsu_u_def, axiom, ![TV_u_27a]: ![V_y, V_x]: s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y)).
fof(ah4s_measures_spaceu_u_def, axiom, ![TV_u_27a]: ![V_y, V_x]: s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y))))) = s(t_fun(TV_u_27a,t_bool),V_x)).
fof(ch4s_probabilitys_PROBu_u_POSITIVE, conjecture, ![TV_u_27a]: ![V_s, V_p]: ((p(s(t_bool,h4s_probabilitys_probu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_probabilitys_events),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_probabilitys_prob),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_p))),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
