%   ORIGINAL: h4/lebesgue/Radon__Nikodym
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/COND__RATOR: !x g f b. h4/bool/COND b f g x = h4/bool/COND b (f x) (g x)
% 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/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/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/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/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% Assm: h4/combin/S__DEF: h4/combin/S = (\f g x. f x (g x))
% Assm: h4/combin/C__DEF: h4/combin/C = (\f x y. f y x)
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/arithmetic/LESS__LESS__EQ__TRANS: !p n m. h4/prim__rec/_3C m n /\ h4/arithmetic/_3C_3D n p ==> h4/prim__rec/_3C m p
% Assm: h4/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c21: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm: h4/numeral/numeral__eq_c3: !n. h4/arithmetic/BIT2 n = h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__lt_c1: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT2 n) <=> T
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/pred__set/IN__ABS: !x P. h4/bool/IN x (\x0. P x0) <=> P x
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pred__set/SUBSET__DEF: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm: h4/pred__set/IN__INTER: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm: h4/pred__set/INTER__SUBSET_c0: !t s. h4/pred__set/SUBSET (h4/pred__set/INTER s t) s
% Assm: h4/pred__set/INTER__SUBSET_c1: !t s. h4/pred__set/SUBSET (h4/pred__set/INTER t s) s
% Assm: h4/pred__set/IN__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = f x /\ h4/bool/IN x s)
% Assm: h4/pred__set/count__def: !n. h4/pred__set/count n = h4/pred__set/GSPEC (\m. h4/pair/_2C m (h4/prim__rec/_3C m n))
% Assm: h4/pred__set/IN__COUNT: !n m. h4/bool/IN m (h4/pred__set/count n) <=> h4/prim__rec/_3C m n
% Assm: h4/pred__set/FINITE__COUNT: !n. h4/pred__set/FINITE (h4/pred__set/count n)
% Assm: h4/real/REAL__ADD__LID: !x. h4/realax/real__add (h4/real/real__of__num h4/num/0) x = x
% Assm: h4/real/REAL__MUL__ASSOC: !z y x. h4/realax/real__mul x (h4/realax/real__mul y z) = h4/realax/real__mul (h4/realax/real__mul x y) z
% Assm: h4/real/REAL__MUL__LINV: !x. ~(x = h4/real/real__of__num h4/num/0) ==> h4/realax/real__mul (h4/realax/inv x) x = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/real/REAL__LT__MUL: !y x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) x /\ h4/realax/real__lt (h4/real/real__of__num h4/num/0) y ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/realax/real__mul x y)
% Assm: h4/real/real__sub0: !y x. h4/real/real__sub x y = h4/realax/real__add x (h4/realax/real__neg y)
% Assm: h4/real/real__div: !y x. h4/real/_2F x y = h4/realax/real__mul x (h4/realax/inv y)
% Assm: h4/real/REAL__MUL__RID: !x. h4/realax/real__mul x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm: h4/real/REAL__LT__LE: !y x. h4/realax/real__lt x y <=> h4/real/real__lte x y /\ ~(x = y)
% Assm: h4/real/REAL__LT__IMP__LE: !y x. h4/realax/real__lt x y ==> h4/real/real__lte x y
% Assm: h4/real/REAL__LTE__TRANS: !z y x. h4/realax/real__lt x y /\ h4/real/real__lte y z ==> h4/realax/real__lt x z
% Assm: h4/real/REAL__LET__TRANS: !z y x. h4/real/real__lte x y /\ h4/realax/real__lt y z ==> h4/realax/real__lt x z
% Assm: h4/real/REAL__LE__ANTISYM: !y x. h4/real/real__lte x y /\ h4/real/real__lte y x <=> x = y
% Assm: h4/real/REAL__LE__MUL: !y x. h4/real/real__lte (h4/real/real__of__num h4/num/0) x /\ h4/real/real__lte (h4/real/real__of__num h4/num/0) y ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/realax/real__mul x y)
% Assm: h4/real/REAL__SUB__REFL: !x. h4/real/real__sub x x = h4/real/real__of__num h4/num/0
% Assm: h4/real/REAL__SUB__0: !y x. h4/real/real__sub x y = h4/real/real__of__num h4/num/0 <=> x = y
% Assm: h4/real/REAL__NEG__0: h4/realax/real__neg (h4/real/real__of__num h4/num/0) = h4/real/real__of__num h4/num/0
% Assm: h4/real/REAL__SUB__LT: !y x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/real__sub x y) <=> h4/realax/real__lt y x
% Assm: h4/real/REAL__SUB__LE: !y x. h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/real/real__sub x y) <=> h4/real/real__lte y x
% Assm: h4/real/REAL__LT__IMP__NE: !y x. h4/realax/real__lt x y ==> ~(x = y)
% Assm: h4/real/REAL__LT__HALF1: !d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/_2F d (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) d
% Assm: h4/real/REAL__SUB__RZERO: !x. h4/real/real__sub x (h4/real/real__of__num h4/num/0) = x
% Assm: h4/real/REAL__INV__MUL: !y x. ~(x = h4/real/real__of__num h4/num/0) /\ ~(y = h4/real/real__of__num h4/num/0) ==> h4/realax/inv (h4/realax/real__mul x y) = h4/realax/real__mul (h4/realax/inv x) (h4/realax/inv y)
% Assm: h4/real/REAL__LT__DIV: !y x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) x /\ h4/realax/real__lt (h4/real/real__of__num h4/num/0) y ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/_2F x y)
% Assm: h4/real/REAL__NEG__HALF: !x. h4/real/real__sub x (h4/real/_2F x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) = h4/real/_2F x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))
% Assm: h4/real/eq__ints_c0: !n m. h4/real/real__of__num n = h4/real/real__of__num m <=> n = m
% Assm: h4/real/lt__int_c0: !n m. h4/realax/real__lt (h4/real/real__of__num n) (h4/real/real__of__num m) <=> h4/prim__rec/_3C n m
% Assm: h4/seq/sums0: !s f. h4/seq/sums f s <=> h4/seq/_2D_2D_3E (\n. h4/real/sum (h4/pair/_2C h4/num/0 n) f) s
% Assm: h4/seq/SER__SUB: !y0 y x0 x. h4/seq/sums x x0 /\ h4/seq/sums y y0 ==> h4/seq/sums (\n. h4/real/real__sub (x n) (y n)) (h4/real/real__sub x0 y0)
% Assm: h4/real__sigma/REAL__SUM__IMAGE__COUNT: !n f. h4/real__sigma/REAL__SUM__IMAGE f (h4/pred__set/count n) = h4/real/sum (h4/pair/_2C h4/num/0 n) 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/util__prob/SUBSET__INTER2: !t s. h4/pred__set/SUBSET s t ==> h4/pred__set/INTER t s = s
% Assm: h4/extreal/extreal__of__num__def: !n. h4/extreal/extreal__of__num n = h4/extreal/Normal (h4/real/real__of__num n)
% Assm: h4/extreal/extreal__sub__def_c0: !y x. h4/extreal/extreal__sub (h4/extreal/Normal x) (h4/extreal/Normal y) = h4/extreal/Normal (h4/real/real__sub x y)
% Assm: h4/extreal/extreal__le__def_c0: !y x. h4/extreal/extreal__le (h4/extreal/Normal x) (h4/extreal/Normal y) <=> h4/real/real__lte x y
% Assm: h4/extreal/extreal__le__def_c0: !y x. h4/extreal/extreal__le (h4/extreal/Normal x) (h4/extreal/Normal y) <=> h4/real/real__lte x y
% Assm: h4/extreal/extreal__le__def_c1: !a. h4/extreal/extreal__le h4/extreal/NegInf a <=> T
% Assm: h4/extreal/extreal__le__def_c2: h4/extreal/extreal__le h4/extreal/PosInf h4/extreal/PosInf <=> T
% Assm: h4/extreal/extreal__le__def_c3: !v2. h4/extreal/extreal__le (h4/extreal/Normal v2) h4/extreal/PosInf <=> T
% Assm: h4/extreal/extreal__le__def_c4: h4/extreal/extreal__le h4/extreal/PosInf h4/extreal/NegInf <=> F
% Assm: h4/extreal/extreal__le__def_c5: !v3. h4/extreal/extreal__le (h4/extreal/Normal v3) h4/extreal/NegInf <=> F
% Assm: h4/extreal/extreal__le__def_c6: !v5. h4/extreal/extreal__le h4/extreal/PosInf (h4/extreal/Normal v5) <=> F
% Assm: h4/extreal/extreal__lt__def: !y x. h4/extreal/extreal__lt x y <=> ~h4/extreal/extreal__le y x
% Assm: h4/extreal/extreal__mul__def_c0: h4/extreal/extreal__mul h4/extreal/NegInf 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_c2: h4/extreal/extreal__mul h4/extreal/PosInf h4/extreal/NegInf = h4/extreal/NegInf
% 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_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_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_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_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_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/mono__increasing__def: !f. h4/extreal/mono__increasing f <=> (!m n. h4/arithmetic/_3C_3D m n ==> h4/real/real__lte (f m) (f n))
% Assm: h4/extreal/extreal__cases: !x. x = h4/extreal/NegInf \/ x = h4/extreal/PosInf \/ (?r. x = h4/extreal/Normal r)
% Assm: h4/extreal/extreal__not__infty_c0: !x. ~(h4/extreal/Normal x = h4/extreal/NegInf)
% Assm: h4/extreal/extreal__not__infty_c1: !x. ~(h4/extreal/Normal x = h4/extreal/PosInf)
% Assm: h4/extreal/num__not__infty_c0: !n. ~(h4/extreal/extreal__of__num n = h4/extreal/NegInf)
% Assm: h4/extreal/extreal__11: !a_27 a. h4/extreal/Normal a = h4/extreal/Normal a_27 <=> a = a_27
% Assm: h4/extreal/mul__rzero: !x. h4/extreal/extreal__mul x (h4/extreal/extreal__of__num h4/num/0) = h4/extreal/extreal__of__num h4/num/0
% Assm: h4/extreal/mul__rone: !x. h4/extreal/extreal__mul x (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm: h4/extreal/extreal__lt__eq: !y x. h4/extreal/extreal__lt (h4/extreal/Normal x) (h4/extreal/Normal y) <=> h4/realax/real__lt x y
% Assm: h4/extreal/le__refl: !x. h4/extreal/extreal__le x x
% Assm: h4/extreal/lt__infty_c0: !y. h4/extreal/extreal__lt h4/extreal/NegInf (h4/extreal/Normal y)
% Assm: h4/extreal/lt__infty_c1: !y. h4/extreal/extreal__lt (h4/extreal/Normal y) h4/extreal/PosInf
% Assm: h4/extreal/lt__infty_c2: h4/extreal/extreal__lt h4/extreal/NegInf h4/extreal/PosInf
% Assm: h4/extreal/lt__infty_c3: !x. ~h4/extreal/extreal__lt x h4/extreal/NegInf
% Assm: h4/extreal/lt__infty_c4: !x. ~h4/extreal/extreal__lt h4/extreal/PosInf x
% Assm: h4/extreal/lt__infty_c5: !x. ~(x = h4/extreal/PosInf) <=> h4/extreal/extreal__lt x h4/extreal/PosInf
% Assm: h4/extreal/lt__infty_c6: !x. ~(x = h4/extreal/NegInf) <=> h4/extreal/extreal__lt h4/extreal/NegInf x
% Assm: h4/extreal/lt__imp__le: !y x. h4/extreal/extreal__lt x y ==> h4/extreal/extreal__le x y
% Assm: h4/extreal/le__trans: !z y x. h4/extreal/extreal__le x y /\ h4/extreal/extreal__le y z ==> h4/extreal/extreal__le x z
% Assm: h4/extreal/let__trans: !z y x. h4/extreal/extreal__le x y /\ h4/extreal/extreal__lt y z ==> h4/extreal/extreal__lt x z
% Assm: h4/extreal/le__antisym: !y x. h4/extreal/extreal__le x y /\ h4/extreal/extreal__le y x <=> x = y
% Assm: h4/extreal/lte__trans: !z y x. h4/extreal/extreal__lt x y /\ h4/extreal/extreal__le y z ==> h4/extreal/extreal__lt x z
% 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__add2: !z y x w. h4/extreal/extreal__le w x /\ h4/extreal/extreal__le y z ==> h4/extreal/extreal__le (h4/extreal/extreal__add w y) (h4/extreal/extreal__add x z)
% Assm: h4/extreal/let__add2: !z y x w. ~(w = h4/extreal/NegInf) /\ ~(w = h4/extreal/PosInf) /\ h4/extreal/extreal__le w x /\ h4/extreal/extreal__lt y z ==> h4/extreal/extreal__lt (h4/extreal/extreal__add w y) (h4/extreal/extreal__add x z)
% Assm: h4/extreal/le__ladd__imp: !z y x. h4/extreal/extreal__le y z ==> h4/extreal/extreal__le (h4/extreal/extreal__add x y) (h4/extreal/extreal__add x z)
% Assm: h4/extreal/add__rzero: !x. h4/extreal/extreal__add x (h4/extreal/extreal__of__num h4/num/0) = x
% Assm: h4/extreal/sub__zero__le: !y x. h4/extreal/extreal__le x y <=> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/extreal/extreal__sub y x)
% Assm: h4/extreal/sub__add2: !y x. ~(x = h4/extreal/NegInf) /\ ~(x = h4/extreal/PosInf) ==> h4/extreal/extreal__add x (h4/extreal/extreal__sub y x) = y
% Assm: h4/extreal/ext__mono__increasing__suc: !f. h4/extreal/ext__mono__increasing f <=> (!n. h4/extreal/extreal__le (f n) (f (h4/num/SUC n)))
% Assm: h4/extreal/EXTREAL__SUM__IMAGE__MONO__SET: !t s f. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t /\ h4/pred__set/SUBSET s t /\ (!x. h4/bool/IN x t ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) ==> h4/extreal/extreal__le (h4/extreal/EXTREAL__SUM__IMAGE f s) (h4/extreal/EXTREAL__SUM__IMAGE f t)
% Assm: h4/extreal/EXTREAL__SUM__IMAGE__NORMAL: !s f. h4/pred__set/FINITE s ==> h4/extreal/EXTREAL__SUM__IMAGE (\x. h4/extreal/Normal (f x)) s = h4/extreal/Normal (h4/real__sigma/REAL__SUM__IMAGE f s)
% Assm: h4/extreal/le__sup__imp: !x p. p x ==> h4/extreal/extreal__le x (h4/extreal/extreal__sup p)
% Assm: h4/extreal/sup__le: !x p. h4/extreal/extreal__le (h4/extreal/extreal__sup p) x <=> (!y. p y ==> h4/extreal/extreal__le y x)
% Assm: h4/extreal/le__sup: !x p. h4/extreal/extreal__le x (h4/extreal/extreal__sup p) <=> (!y. (!z. p z ==> h4/extreal/extreal__le z y) ==> h4/extreal/extreal__le x y)
% Assm: h4/extreal/sup__const: !x. h4/extreal/extreal__sup (\y. y = x) = x
% Assm: h4/extreal/sup__seq: !l f. h4/extreal/mono__increasing f ==> (h4/seq/_2D_2D_3E f l <=> h4/extreal/extreal__sup (h4/pred__set/IMAGE (\n. h4/extreal/Normal (f n)) h4/pred__set/UNIV) = h4/extreal/Normal l)
% Assm: h4/extreal/ext__suminf__def: !f. h4/extreal/ext__suminf f = h4/extreal/extreal__sup (h4/pred__set/IMAGE (\n. h4/extreal/EXTREAL__SUM__IMAGE f (h4/pred__set/count n)) h4/pred__set/UNIV)
% Assm: h4/extreal/ext__suminf__cmul: !f c. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) c /\ (!n. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f n)) ==> h4/extreal/ext__suminf (\n. h4/extreal/extreal__mul c (f n)) = h4/extreal/extreal__mul c (h4/extreal/ext__suminf f)
% Assm: h4/measure/space__def: !y x. h4/measure/space (h4/pair/_2C x y) = x
% Assm: h4/measure/subsets__def: !y x. h4/measure/subsets (h4/pair/_2C x y) = y
% Assm: h4/measure/m__space__def: !sts sp mu. h4/measure/m__space (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sp
% Assm: h4/measure/measurable__sets__def: !sts sp mu. h4/measure/measurable__sets (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sts
% Assm: h4/measure/measure__def: !sts sp mu. h4/measure/measure (h4/pair/_2C sp (h4/pair/_2C sts mu)) = mu
% Assm: h4/measure/positive__def: !m. h4/measure/positive m <=> h4/measure/measure m h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0 /\ (!s. h4/bool/IN s (h4/measure/measurable__sets m) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/measure/measure m s))
% Assm: h4/measure/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/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/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/measure/INCREASING: !t s m. h4/measure/increasing m /\ h4/pred__set/SUBSET s t /\ h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/bool/IN t (h4/measure/measurable__sets m) ==> h4/real/real__lte (h4/measure/measure m s) (h4/measure/measure m t)
% Assm: h4/measure/MEASURE__EMPTY: !m. h4/measure/measure__space m ==> h4/measure/measure m h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0
% Assm: h4/measure/MEASURE__SPACE__INCREASING: !m. h4/measure/measure__space m ==> h4/measure/increasing m
% Assm: h4/measure/MEASURE__SPACE__POSITIVE: !m. h4/measure/measure__space m ==> h4/measure/positive m
% Assm: h4/measure/MEASURE__SPACE__INTER: !t s m. h4/measure/measure__space m /\ h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/bool/IN t (h4/measure/measurable__sets m) ==> h4/bool/IN (h4/pred__set/INTER s t) (h4/measure/measurable__sets m)
% Assm: h4/measure/MEASURE__SPACE__SUBSET__MSPACE: !m A. h4/measure/measure__space m /\ h4/bool/IN A (h4/measure/measurable__sets m) ==> h4/pred__set/SUBSET A (h4/measure/m__space m)
% Assm: h4/measure/MEASURE__SPACE__EMPTY__MEASURABLE: !m. h4/measure/measure__space m ==> h4/bool/IN h4/pred__set/EMPTY (h4/measure/measurable__sets m)
% Assm: h4/measure/MEASURE__SPACE__MSPACE__MEASURABLE: !m. h4/measure/measure__space m ==> h4/bool/IN (h4/measure/m__space m) (h4/measure/measurable__sets m)
% Assm: h4/measure/MEASURE__SPACE__CMUL: !m c. h4/measure/measure__space m /\ h4/real/real__lte (h4/real/real__of__num h4/num/0) c ==> h4/measure/measure__space (h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/measure/measurable__sets m) (\a. h4/realax/real__mul c (h4/measure/measure m a))))
% Assm: h4/measure/IN__MEASURABLE__BOREL__INDICATOR: !f a A. h4/measure/sigma__algebra a /\ h4/bool/IN A (h4/measure/subsets a) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> f x = h4/measure/indicator__fn A x) ==> h4/bool/IN f (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/measure/IN__MEASURABLE__BOREL__CMUL: !z g f a. h4/measure/sigma__algebra a /\ h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> g x = h4/extreal/extreal__mul (h4/extreal/Normal z) (f x)) ==> h4/bool/IN g (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/measure/IN__MEASURABLE__BOREL__ADD: !h g f a. h4/measure/sigma__algebra a /\ h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) /\ h4/bool/IN g (h4/measure/measurable a h4/measure/Borel) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> h x = h4/extreal/extreal__add (f x) (g x)) ==> h4/bool/IN h (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/measure/IN__MEASURABLE__BOREL__MUL__INDICATOR: !s f a. h4/measure/sigma__algebra a /\ h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) /\ h4/bool/IN s (h4/measure/subsets a) ==> h4/bool/IN (\x. h4/extreal/extreal__mul (f x) (h4/measure/indicator__fn s x)) (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/measure/IN__MEASURABLE__BOREL__MONO__SUP: !fn f a. h4/measure/sigma__algebra a /\ (!n. h4/bool/IN (fn n) (h4/measure/measurable a h4/measure/Borel)) /\ (!n x. h4/bool/IN x (h4/measure/space a) ==> h4/extreal/extreal__le (fn n x) (fn (h4/num/SUC n) x)) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> f x = h4/extreal/extreal__sup (h4/pred__set/IMAGE (\n. fn n x) h4/pred__set/UNIV)) ==> h4/bool/IN f (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/measure/indicator__fn__suminf: !x a. (!m n. ~(m = n) ==> h4/pred__set/DISJOINT (a m) (a n)) ==> h4/extreal/ext__suminf (\i. h4/measure/indicator__fn (a i) x) = h4/measure/indicator__fn (h4/pred__set/BIGUNION (h4/pred__set/IMAGE a h4/pred__set/UNIV)) x
% Assm: h4/lebesgue/pos__fn__integral__mspace: !m f. h4/measure/measure__space m /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) ==> h4/lebesgue/pos__fn__integral m f = h4/lebesgue/pos__fn__integral m (\x. h4/extreal/extreal__mul (f x) (h4/measure/indicator__fn (h4/measure/m__space m) x))
% Assm: h4/lebesgue/pos__fn__integral__zero: !m. h4/measure/measure__space m ==> h4/lebesgue/pos__fn__integral m (\x. h4/extreal/extreal__of__num h4/num/0) = h4/extreal/extreal__of__num h4/num/0
% Assm: h4/lebesgue/pos__fn__integral__pos: !m f. h4/measure/measure__space m /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/lebesgue/pos__fn__integral m f)
% 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/lebesgue/pos__fn__integral__indicator: !s m. h4/measure/measure__space m /\ h4/bool/IN s (h4/measure/measurable__sets m) ==> h4/lebesgue/pos__fn__integral m (h4/measure/indicator__fn s) = h4/extreal/Normal (h4/measure/measure m s)
% Assm: h4/lebesgue/lebesgue__monotone__convergence: !m fi f. h4/measure/measure__space m /\ (!i. h4/bool/IN (fi i) (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (fi i x)) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) /\ (!x. h4/extreal/ext__mono__increasing (\i. fi i x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. fi i x) h4/pred__set/UNIV) = f x) ==> h4/lebesgue/pos__fn__integral m f = h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. h4/lebesgue/pos__fn__integral m (fi i)) h4/pred__set/UNIV)
% Assm: h4/lebesgue/lebesgue__monotone__convergence__subset: !m fi f A. h4/measure/measure__space m /\ (!i. h4/bool/IN (fi i) (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (fi i x)) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. fi i x) h4/pred__set/UNIV) = f x) /\ (!x. h4/extreal/ext__mono__increasing (\i. fi i x)) /\ h4/bool/IN A (h4/measure/measurable__sets m) ==> h4/lebesgue/pos__fn__integral m (\x. h4/extreal/extreal__mul (f x) (h4/measure/indicator__fn A x)) = h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. h4/lebesgue/pos__fn__integral m (\x. h4/extreal/extreal__mul (fi i x) (h4/measure/indicator__fn A x))) h4/pred__set/UNIV)
% Assm: h4/lebesgue/pos__fn__integral__add: !m g f. h4/measure/measure__space m /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x) /\ h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (g x)) /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) /\ h4/bool/IN g (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> h4/lebesgue/pos__fn__integral m (\x. h4/extreal/extreal__add (f x) (g x)) = h4/extreal/extreal__add (h4/lebesgue/pos__fn__integral m f) (h4/lebesgue/pos__fn__integral m g)
% Assm: h4/lebesgue/pos__fn__integral__suminf: !m f. h4/measure/measure__space m /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f i x)) /\ (!i. h4/bool/IN (f i) (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)) ==> h4/lebesgue/pos__fn__integral m (\x. h4/extreal/ext__suminf (\i. f i x)) = h4/extreal/ext__suminf (\i. h4/lebesgue/pos__fn__integral m (f i))
% Assm: h4/lebesgue/RADON__F__def: !v m. h4/lebesgue/RADON__F m v = h4/pred__set/GSPEC (\f. h4/pair/_2C f (h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) /\ (!A. h4/bool/IN A (h4/measure/measurable__sets m) ==> h4/extreal/extreal__le (h4/lebesgue/pos__fn__integral m (\x. h4/extreal/extreal__mul (f x) (h4/measure/indicator__fn A x))) (h4/extreal/Normal (h4/measure/measure v A)))))
% Assm: h4/lebesgue/RADON__F__integrals__def: !v m. h4/lebesgue/RADON__F__integrals m v = h4/pred__set/GSPEC (\r. h4/pair/_2C r (?f. r = h4/lebesgue/pos__fn__integral m f /\ h4/bool/IN f (h4/lebesgue/RADON__F m v)))
% Assm: h4/lebesgue/measure__absolutely__continuous__def: !v m. h4/lebesgue/measure__absolutely__continuous m v <=> (!A. h4/bool/IN A (h4/measure/measurable__sets m) /\ h4/measure/measure v A = h4/real/real__of__num h4/num/0 ==> h4/measure/measure m A = h4/real/real__of__num h4/num/0)
% Assm: h4/lebesgue/lemma__radon__seq__conv__sup: !v m. h4/measure/measure__space m /\ h4/measure/measure__space v /\ h4/measure/m__space v = h4/measure/m__space m /\ h4/measure/measurable__sets v = h4/measure/measurable__sets m ==> (?f__n. (!n. h4/bool/IN (f__n n) (h4/lebesgue/RADON__F m v)) /\ (!x n. h4/extreal/extreal__le (f__n n x) (f__n (h4/num/SUC n) x)) /\ h4/extreal/extreal__sup (h4/pred__set/IMAGE (\n. h4/lebesgue/pos__fn__integral m (f__n n)) h4/pred__set/UNIV) = h4/extreal/extreal__sup (h4/lebesgue/RADON__F__integrals m v))
% Assm: h4/lebesgue/RN__lemma2: !v m. h4/measure/measure__space m /\ h4/measure/measure__space v /\ h4/measure/m__space v = h4/measure/m__space m /\ h4/measure/measurable__sets v = h4/measure/measurable__sets m ==> (?A. h4/bool/IN A (h4/measure/measurable__sets m) /\ h4/real/real__lte (h4/real/real__sub (h4/measure/measure m (h4/measure/m__space m)) (h4/measure/measure v (h4/measure/m__space m))) (h4/real/real__sub (h4/measure/measure m A) (h4/measure/measure v A)) /\ (!B. h4/bool/IN B (h4/measure/measurable__sets m) /\ h4/pred__set/SUBSET B A ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/real/real__sub (h4/measure/measure m B) (h4/measure/measure v B))))
% Goal: !v m. h4/measure/measure__space m /\ h4/measure/measure__space v /\ h4/measure/m__space v = h4/measure/m__space m /\ h4/measure/measurable__sets v = h4/measure/measurable__sets m /\ h4/lebesgue/measure__absolutely__continuous v m ==> (?f. h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) /\ (!A. h4/bool/IN A (h4/measure/measurable__sets m) ==> h4/lebesgue/pos__fn__integral m (\x. h4/extreal/extreal__mul (f x) (h4/measure/indicator__fn A x)) = h4/extreal/Normal (h4/measure/measure v A)))
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_ORu_u_DEF]: !x x'. $or x x' <=> (!t. (x ==> t) ==> (x' ==> t) ==> t)
% Assm [h4s_bools_Fu_u_DEF]: F <=> (!t. t)
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_CONDu_u_RATOR]: !x g f b. happ (h4/bool/COND b f g) x = h4/bool/COND b (happ f x) (happ g x)
% 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_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_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_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_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_markers_Abbrevu_u_def]: !x. h4/marker/Abbrev x <=> x
% Assm [h4s_combins_Su_u_DEF]: !x x x. h4/combin/S x x x = happ (happ x x) (happ x x)
% Assm [h4s_combins_Cu_u_DEF]: !x x x. h4/combin/C x x x = happ (happ x x) x
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_combins_Iu_u_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% Assm [h4s_pairs_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_arithmetics_LESSu_u_LESSu_u_EQu_u_TRANS]: !p n m. h4/prim__rec/_3C m n /\ h4/arithmetic/_3C_3D n p ==> h4/prim__rec/_3C m p
% Assm [h4s_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c21]: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm [h4s_numerals_numeralu_u_equ_c3]: !n. h4/arithmetic/BIT2 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_ltu_c1]: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT2 n) <=> T
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_predu_u_sets_INu_u_ABS]: !_0. (!P x0. happ (happ _0 P) x0 <=> happ P x0) ==> (!x P. h4/bool/IN x (happ _0 P) <=> happ P x)
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_SUBSETu_u_DEF]: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_INu_u_INTER]: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm [h4s_predu_u_sets_INTERu_u_SUBSETu_c0]: !t s. h4/pred__set/SUBSET (h4/pred__set/INTER s t) s
% Assm [h4s_predu_u_sets_INTERu_u_SUBSETu_c1]: !t s. h4/pred__set/SUBSET (h4/pred__set/INTER t s) s
% Assm [h4s_predu_u_sets_INu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = happ f x /\ h4/bool/IN x s)
% Assm [h4s_predu_u_sets_countu_u_def]: !_0. (!n m. happ (happ _0 n) m = h4/pair/_2C m (h4/prim__rec/_3C m n)) ==> (!n. h4/pred__set/count n = h4/pred__set/GSPEC (happ _0 n))
% Assm [h4s_predu_u_sets_INu_u_COUNT]: !n m. h4/bool/IN m (h4/pred__set/count n) <=> h4/prim__rec/_3C m n
% Assm [h4s_predu_u_sets_FINITEu_u_COUNT]: !n. h4/pred__set/FINITE (h4/pred__set/count n)
% Assm [h4s_reals_REALu_u_ADDu_u_LID]: !x. h4/realax/real__add (h4/real/real__of__num h4/num/0) x = x
% Assm [h4s_reals_REALu_u_MULu_u_ASSOC]: !z y x. h4/realax/real__mul x (h4/realax/real__mul y z) = h4/realax/real__mul (h4/realax/real__mul x y) z
% Assm [h4s_reals_REALu_u_MULu_u_LINV]: !x. ~(x = h4/real/real__of__num h4/num/0) ==> h4/realax/real__mul (h4/realax/inv x) x = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_reals_REALu_u_LTu_u_MUL]: !y x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) x /\ h4/realax/real__lt (h4/real/real__of__num h4/num/0) y ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/realax/real__mul x y)
% Assm [h4s_reals_realu_u_sub0]: !y x. h4/real/real__sub x y = h4/realax/real__add x (h4/realax/real__neg y)
% Assm [h4s_reals_realu_u_div]: !y x. h4/real/_2F x y = h4/realax/real__mul x (h4/realax/inv y)
% Assm [h4s_reals_REALu_u_MULu_u_RID]: !x. h4/realax/real__mul x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm [h4s_reals_REALu_u_LTu_u_LE]: !y x. h4/realax/real__lt x y <=> h4/real/real__lte x y /\ ~(x = y)
% Assm [h4s_reals_REALu_u_LTu_u_IMPu_u_LE]: !y x. h4/realax/real__lt x y ==> h4/real/real__lte x y
% Assm [h4s_reals_REALu_u_LTEu_u_TRANS]: !z y x. h4/realax/real__lt x y /\ h4/real/real__lte y z ==> h4/realax/real__lt x z
% Assm [h4s_reals_REALu_u_LETu_u_TRANS]: !z y x. h4/real/real__lte x y /\ h4/realax/real__lt y z ==> h4/realax/real__lt x z
% Assm [h4s_reals_REALu_u_LEu_u_ANTISYM]: !y x. h4/real/real__lte x y /\ h4/real/real__lte y x <=> x = y
% Assm [h4s_reals_REALu_u_LEu_u_MUL]: !y x. h4/real/real__lte (h4/real/real__of__num h4/num/0) x /\ h4/real/real__lte (h4/real/real__of__num h4/num/0) y ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/realax/real__mul x y)
% Assm [h4s_reals_REALu_u_SUBu_u_REFL]: !x. h4/real/real__sub x x = h4/real/real__of__num h4/num/0
% Assm [h4s_reals_REALu_u_SUBu_u_0]: !y x. h4/real/real__sub x y = h4/real/real__of__num h4/num/0 <=> x = y
% Assm [h4s_reals_REALu_u_NEGu_u_0]: h4/realax/real__neg (h4/real/real__of__num h4/num/0) = h4/real/real__of__num h4/num/0
% Assm [h4s_reals_REALu_u_SUBu_u_LT]: !y x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/real__sub x y) <=> h4/realax/real__lt y x
% Assm [h4s_reals_REALu_u_SUBu_u_LE]: !y x. h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/real/real__sub x y) <=> h4/real/real__lte y x
% Assm [h4s_reals_REALu_u_LTu_u_IMPu_u_NE]: !y x. h4/realax/real__lt x y ==> ~(x = y)
% Assm [h4s_reals_REALu_u_LTu_u_HALF1]: !d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/_2F d (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) d
% Assm [h4s_reals_REALu_u_SUBu_u_RZERO]: !x. h4/real/real__sub x (h4/real/real__of__num h4/num/0) = x
% Assm [h4s_reals_REALu_u_INVu_u_MUL]: !y x. ~(x = h4/real/real__of__num h4/num/0) /\ ~(y = h4/real/real__of__num h4/num/0) ==> h4/realax/inv (h4/realax/real__mul x y) = h4/realax/real__mul (h4/realax/inv x) (h4/realax/inv y)
% Assm [h4s_reals_REALu_u_LTu_u_DIV]: !y x. h4/realax/real__lt (h4/real/real__of__num h4/num/0) x /\ h4/realax/real__lt (h4/real/real__of__num h4/num/0) y ==> h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/_2F x y)
% Assm [h4s_reals_REALu_u_NEGu_u_HALF]: !x. h4/real/real__sub x (h4/real/_2F x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) = h4/real/_2F x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))
% Assm [h4s_reals_equ_u_intsu_c0]: !n m. h4/real/real__of__num n = h4/real/real__of__num m <=> n = m
% Assm [h4s_reals_ltu_u_intu_c0]: !n m. h4/realax/real__lt (h4/real/real__of__num n) (h4/real/real__of__num m) <=> h4/prim__rec/_3C n m
% Assm [h4s_seqs_sums0]: !_0. (!f n. happ (happ _0 f) n = h4/real/sum (h4/pair/_2C h4/num/0 n) f) ==> (!s f. h4/seq/sums f s <=> h4/seq/_2D_2D_3E (happ _0 f) s)
% Assm [h4s_seqs_SERu_u_SUB]: !_0. (!x y n. happ (happ (happ _0 x) y) n = h4/real/real__sub (happ x n) (happ y n)) ==> (!y0 y x0 x. h4/seq/sums x x0 /\ h4/seq/sums y y0 ==> h4/seq/sums (happ (happ _0 x) y) (h4/real/real__sub x0 y0))
% Assm [h4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_COUNT]: !n f. h4/real__sigma/REAL__SUM__IMAGE f (h4/pred__set/count n) = h4/real/sum (h4/pair/_2C h4/num/0 n) 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_utilu_u_probs_SUBSETu_u_INTER2]: !t s. h4/pred__set/SUBSET s t ==> h4/pred__set/INTER t s = s
% Assm [h4s_extreals_extrealu_u_ofu_u_numu_u_def]: !n. h4/extreal/extreal__of__num n = h4/extreal/Normal (h4/real/real__of__num n)
% Assm [h4s_extreals_extrealu_u_subu_u_defu_c0]: !y x. h4/extreal/extreal__sub (h4/extreal/Normal x) (h4/extreal/Normal y) = h4/extreal/Normal (h4/real/real__sub x y)
% Assm [h4s_extreals_extrealu_u_leu_u_defu_c0]: !y x. h4/extreal/extreal__le (h4/extreal/Normal x) (h4/extreal/Normal y) <=> h4/real/real__lte x y
% Assm [h4s_extreals_extrealu_u_leu_u_defu_c0]: !y x. h4/extreal/extreal__le (h4/extreal/Normal x) (h4/extreal/Normal y) <=> h4/real/real__lte x y
% Assm [h4s_extreals_extrealu_u_leu_u_defu_c1]: !a. h4/extreal/extreal__le h4/extreal/NegInf a <=> T
% Assm [h4s_extreals_extrealu_u_leu_u_defu_c2]: h4/extreal/extreal__le h4/extreal/PosInf h4/extreal/PosInf <=> T
% Assm [h4s_extreals_extrealu_u_leu_u_defu_c3]: !v2. h4/extreal/extreal__le (h4/extreal/Normal v2) h4/extreal/PosInf <=> T
% Assm [h4s_extreals_extrealu_u_leu_u_defu_c4]: h4/extreal/extreal__le h4/extreal/PosInf h4/extreal/NegInf <=> F
% Assm [h4s_extreals_extrealu_u_leu_u_defu_c5]: !v3. h4/extreal/extreal__le (h4/extreal/Normal v3) h4/extreal/NegInf <=> F
% Assm [h4s_extreals_extrealu_u_leu_u_defu_c6]: !v5. h4/extreal/extreal__le h4/extreal/PosInf (h4/extreal/Normal v5) <=> F
% Assm [h4s_extreals_extrealu_u_ltu_u_def]: !y x. h4/extreal/extreal__lt x y <=> ~h4/extreal/extreal__le y x
% Assm [h4s_extreals_extrealu_u_mulu_u_defu_c0]: h4/extreal/extreal__mul h4/extreal/NegInf 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_c2]: h4/extreal/extreal__mul h4/extreal/PosInf h4/extreal/NegInf = h4/extreal/NegInf
% 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_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_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_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_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_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_monou_u_increasingu_u_def]: !f. h4/extreal/mono__increasing f <=> (!m n. h4/arithmetic/_3C_3D m n ==> h4/real/real__lte (happ f m) (happ f n))
% Assm [h4s_extreals_extrealu_u_cases]: !x. x = h4/extreal/NegInf \/ x = h4/extreal/PosInf \/ (?r. x = h4/extreal/Normal r)
% Assm [h4s_extreals_extrealu_u_notu_u_inftyu_c0]: !x. ~(h4/extreal/Normal x = h4/extreal/NegInf)
% Assm [h4s_extreals_extrealu_u_notu_u_inftyu_c1]: !x. ~(h4/extreal/Normal x = h4/extreal/PosInf)
% Assm [h4s_extreals_numu_u_notu_u_inftyu_c0]: !n. ~(h4/extreal/extreal__of__num n = h4/extreal/NegInf)
% Assm [h4s_extreals_extrealu_u_11]: !a_27 a. h4/extreal/Normal a = h4/extreal/Normal a_27 <=> a = a_27
% Assm [h4s_extreals_mulu_u_rzero]: !x. h4/extreal/extreal__mul x (h4/extreal/extreal__of__num h4/num/0) = h4/extreal/extreal__of__num h4/num/0
% Assm [h4s_extreals_mulu_u_rone]: !x. h4/extreal/extreal__mul x (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm [h4s_extreals_extrealu_u_ltu_u_eq]: !y x. h4/extreal/extreal__lt (h4/extreal/Normal x) (h4/extreal/Normal y) <=> h4/realax/real__lt x y
% Assm [h4s_extreals_leu_u_refl]: !x. h4/extreal/extreal__le x x
% Assm [h4s_extreals_ltu_u_inftyu_c0]: !y. h4/extreal/extreal__lt h4/extreal/NegInf (h4/extreal/Normal y)
% Assm [h4s_extreals_ltu_u_inftyu_c1]: !y. h4/extreal/extreal__lt (h4/extreal/Normal y) h4/extreal/PosInf
% Assm [h4s_extreals_ltu_u_inftyu_c2]: h4/extreal/extreal__lt h4/extreal/NegInf h4/extreal/PosInf
% Assm [h4s_extreals_ltu_u_inftyu_c3]: !x. ~h4/extreal/extreal__lt x h4/extreal/NegInf
% Assm [h4s_extreals_ltu_u_inftyu_c4]: !x. ~h4/extreal/extreal__lt h4/extreal/PosInf x
% Assm [h4s_extreals_ltu_u_inftyu_c5]: !x. ~(x = h4/extreal/PosInf) <=> h4/extreal/extreal__lt x h4/extreal/PosInf
% Assm [h4s_extreals_ltu_u_inftyu_c6]: !x. ~(x = h4/extreal/NegInf) <=> h4/extreal/extreal__lt h4/extreal/NegInf x
% Assm [h4s_extreals_ltu_u_impu_u_le]: !y x. h4/extreal/extreal__lt x y ==> h4/extreal/extreal__le x y
% Assm [h4s_extreals_leu_u_trans]: !z y x. h4/extreal/extreal__le x y /\ h4/extreal/extreal__le y z ==> h4/extreal/extreal__le x z
% Assm [h4s_extreals_letu_u_trans]: !z y x. h4/extreal/extreal__le x y /\ h4/extreal/extreal__lt y z ==> h4/extreal/extreal__lt x z
% Assm [h4s_extreals_leu_u_antisym]: !y x. h4/extreal/extreal__le x y /\ h4/extreal/extreal__le y x <=> x = y
% Assm [h4s_extreals_lteu_u_trans]: !z y x. h4/extreal/extreal__lt x y /\ h4/extreal/extreal__le y z ==> h4/extreal/extreal__lt x z
% 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_add2]: !z y x w. h4/extreal/extreal__le w x /\ h4/extreal/extreal__le y z ==> h4/extreal/extreal__le (h4/extreal/extreal__add w y) (h4/extreal/extreal__add x z)
% Assm [h4s_extreals_letu_u_add2]: !z y x w. ~(w = h4/extreal/NegInf) /\ ~(w = h4/extreal/PosInf) /\ h4/extreal/extreal__le w x /\ h4/extreal/extreal__lt y z ==> h4/extreal/extreal__lt (h4/extreal/extreal__add w y) (h4/extreal/extreal__add x z)
% Assm [h4s_extreals_leu_u_laddu_u_imp]: !z y x. h4/extreal/extreal__le y z ==> h4/extreal/extreal__le (h4/extreal/extreal__add x y) (h4/extreal/extreal__add x z)
% Assm [h4s_extreals_addu_u_rzero]: !x. h4/extreal/extreal__add x (h4/extreal/extreal__of__num h4/num/0) = x
% Assm [h4s_extreals_subu_u_zerou_u_le]: !y x. h4/extreal/extreal__le x y <=> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/extreal/extreal__sub y x)
% Assm [h4s_extreals_subu_u_add2]: !y x. ~(x = h4/extreal/NegInf) /\ ~(x = h4/extreal/PosInf) ==> h4/extreal/extreal__add x (h4/extreal/extreal__sub y x) = y
% Assm [h4s_extreals_extu_u_monou_u_increasingu_u_suc]: !f. h4/extreal/ext__mono__increasing f <=> (!n. h4/extreal/extreal__le (happ f n) (happ f (h4/num/SUC n)))
% Assm [h4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_MONOu_u_SET]: !t s f. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t /\ h4/pred__set/SUBSET s t /\ (!x. h4/bool/IN x t ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) ==> h4/extreal/extreal__le (h4/extreal/EXTREAL__SUM__IMAGE f s) (h4/extreal/EXTREAL__SUM__IMAGE f t)
% Assm [h4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_NORMAL]: !_0. (!f x. happ (happ _0 f) x = h4/extreal/Normal (happ f x)) ==> (!s f. h4/pred__set/FINITE s ==> h4/extreal/EXTREAL__SUM__IMAGE (happ _0 f) s = h4/extreal/Normal (h4/real__sigma/REAL__SUM__IMAGE f s))
% Assm [h4s_extreals_leu_u_supu_u_imp]: !x p. happ p x ==> h4/extreal/extreal__le x (h4/extreal/extreal__sup p)
% Assm [h4s_extreals_supu_u_le]: !x p. h4/extreal/extreal__le (h4/extreal/extreal__sup p) x <=> (!y. happ p y ==> h4/extreal/extreal__le y x)
% Assm [h4s_extreals_leu_u_sup]: !x p. h4/extreal/extreal__le x (h4/extreal/extreal__sup p) <=> (!y. (!z. happ p z ==> h4/extreal/extreal__le z y) ==> h4/extreal/extreal__le x y)
% Assm [h4s_extreals_supu_u_const]: !_0. (!x y. happ (happ _0 x) y <=> y = x) ==> (!x. h4/extreal/extreal__sup (happ _0 x) = x)
% Assm [h4s_extreals_supu_u_seq]: !_0. (!f n. happ (happ _0 f) n = h4/extreal/Normal (happ f n)) ==> (!l f. h4/extreal/mono__increasing f ==> (h4/seq/_2D_2D_3E f l <=> h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ _0 f) h4/pred__set/UNIV) = h4/extreal/Normal l))
% Assm [h4s_extreals_extu_u_suminfu_u_def]: !_0. (!f n. happ (happ _0 f) n = h4/extreal/EXTREAL__SUM__IMAGE f (h4/pred__set/count n)) ==> (!f. h4/extreal/ext__suminf f = h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ _0 f) h4/pred__set/UNIV))
% Assm [h4s_extreals_extu_u_suminfu_u_cmul]: !_0. (!c f n. happ (happ (happ _0 c) f) n = h4/extreal/extreal__mul c (happ f n)) ==> (!f c. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) c /\ (!n. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f n)) ==> h4/extreal/ext__suminf (happ (happ _0 c) f) = h4/extreal/extreal__mul c (h4/extreal/ext__suminf f))
% Assm [h4s_measures_spaceu_u_def]: !y x. h4/measure/space (h4/pair/_2C x y) = x
% Assm [h4s_measures_subsetsu_u_def]: !y x. h4/measure/subsets (h4/pair/_2C x y) = y
% Assm [h4s_measures_mu_u_spaceu_u_def]: !sts sp mu. h4/measure/m__space (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sp
% Assm [h4s_measures_measurableu_u_setsu_u_def]: !sts sp mu. h4/measure/measurable__sets (h4/pair/_2C sp (h4/pair/_2C sts mu)) = sts
% Assm [h4s_measures_measureu_u_def]: !sts sp mu. h4/measure/measure (h4/pair/_2C sp (h4/pair/_2C sts mu)) = mu
% Assm [h4s_measures_positiveu_u_def]: !m. h4/measure/positive m <=> happ (h4/measure/measure m) h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0 /\ (!s. h4/bool/IN s (h4/measure/measurable__sets m) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ (h4/measure/measure m) s))
% Assm [h4s_measures_countablyu_u_additiveu_u_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 (happ f m0) (happ 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) (happ (h4/measure/measure m) (h4/pred__set/BIGUNION (h4/pred__set/IMAGE f h4/pred__set/UNIV))))
% Assm [h4s_measures_measureu_u_spaceu_u_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 [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_measures_INCREASING]: !t s m. h4/measure/increasing m /\ h4/pred__set/SUBSET s t /\ h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/bool/IN t (h4/measure/measurable__sets m) ==> h4/real/real__lte (happ (h4/measure/measure m) s) (happ (h4/measure/measure m) t)
% Assm [h4s_measures_MEASUREu_u_EMPTY]: !m. h4/measure/measure__space m ==> happ (h4/measure/measure m) h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_INCREASING]: !m. h4/measure/measure__space m ==> h4/measure/increasing m
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_POSITIVE]: !m. h4/measure/measure__space m ==> h4/measure/positive m
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_INTER]: !t s m. h4/measure/measure__space m /\ h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/bool/IN t (h4/measure/measurable__sets m) ==> h4/bool/IN (h4/pred__set/INTER s t) (h4/measure/measurable__sets m)
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_SUBSETu_u_MSPACE]: !m A. h4/measure/measure__space m /\ h4/bool/IN A (h4/measure/measurable__sets m) ==> h4/pred__set/SUBSET A (h4/measure/m__space m)
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_EMPTYu_u_MEASURABLE]: !m. h4/measure/measure__space m ==> h4/bool/IN h4/pred__set/EMPTY (h4/measure/measurable__sets m)
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_MSPACEu_u_MEASURABLE]: !m. h4/measure/measure__space m ==> h4/bool/IN (h4/measure/m__space m) (h4/measure/measurable__sets m)
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_CMUL]: !_0. (!c m a. happ (happ (happ _0 c) m) a = h4/realax/real__mul c (happ (h4/measure/measure m) a)) ==> (!m c. h4/measure/measure__space m /\ h4/real/real__lte (h4/real/real__of__num h4/num/0) c ==> h4/measure/measure__space (h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/measure/measurable__sets m) (happ (happ _0 c) m))))
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_INDICATOR]: !f a A. h4/measure/sigma__algebra a /\ h4/bool/IN A (h4/measure/subsets a) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> happ f x = happ (h4/measure/indicator__fn A) x) ==> h4/bool/IN f (h4/measure/measurable a h4/measure/Borel)
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_CMUL]: !z g f a. h4/measure/sigma__algebra a /\ h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> happ g x = h4/extreal/extreal__mul (h4/extreal/Normal z) (happ f x)) ==> h4/bool/IN g (h4/measure/measurable a h4/measure/Borel)
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_ADD]: !h g f a. h4/measure/sigma__algebra a /\ h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) /\ h4/bool/IN g (h4/measure/measurable a h4/measure/Borel) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> happ h x = h4/extreal/extreal__add (happ f x) (happ g x)) ==> h4/bool/IN h (h4/measure/measurable a h4/measure/Borel)
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_MULu_u_INDICATOR]: !_0. (!f s x. happ (happ (happ _0 f) s) x = h4/extreal/extreal__mul (happ f x) (happ (h4/measure/indicator__fn s) x)) ==> (!s f a. h4/measure/sigma__algebra a /\ h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) /\ h4/bool/IN s (h4/measure/subsets a) ==> h4/bool/IN (happ (happ _0 f) s) (h4/measure/measurable a h4/measure/Borel))
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_MONOu_u_SUP]: !_0. (!fn x n. happ (happ (happ _0 fn) x) n = happ (happ fn n) x) ==> (!fn f a. h4/measure/sigma__algebra a /\ (!n. h4/bool/IN (happ fn n) (h4/measure/measurable a h4/measure/Borel)) /\ (!n x. h4/bool/IN x (h4/measure/space a) ==> h4/extreal/extreal__le (happ (happ fn n) x) (happ (happ fn (h4/num/SUC n)) x)) /\ (!x. h4/bool/IN x (h4/measure/space a) ==> happ f x = h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 fn) x) h4/pred__set/UNIV)) ==> h4/bool/IN f (h4/measure/measurable a h4/measure/Borel))
% Assm [h4s_measures_indicatoru_u_fnu_u_suminf]: !_0. (!a x i. happ (happ (happ _0 a) x) i = happ (h4/measure/indicator__fn (happ a i)) x) ==> (!x a. (!m n. ~(m = n) ==> h4/pred__set/DISJOINT (happ a m) (happ a n)) ==> h4/extreal/ext__suminf (happ (happ _0 a) x) = happ (h4/measure/indicator__fn (h4/pred__set/BIGUNION (h4/pred__set/IMAGE a h4/pred__set/UNIV))) x)
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_mspace]: !_0. (!f m x. happ (happ (happ _0 f) m) x = h4/extreal/extreal__mul (happ f x) (happ (h4/measure/indicator__fn (h4/measure/m__space m)) x)) ==> (!m f. h4/measure/measure__space m /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) ==> h4/lebesgue/pos__fn__integral m f = h4/lebesgue/pos__fn__integral m (happ (happ _0 f) m))
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_zero]: !_0. (!x. happ _0 x = h4/extreal/extreal__of__num h4/num/0) ==> (!m. h4/measure/measure__space m ==> h4/lebesgue/pos__fn__integral m _0 = h4/extreal/extreal__of__num h4/num/0)
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_pos]: !m f. h4/measure/measure__space m /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) ==> h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/lebesgue/pos__fn__integral m f)
% 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 (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_lebesgues_posu_u_fnu_u_integralu_u_indicator]: !s m. h4/measure/measure__space m /\ h4/bool/IN s (h4/measure/measurable__sets m) ==> h4/lebesgue/pos__fn__integral m (h4/measure/indicator__fn s) = h4/extreal/Normal (happ (h4/measure/measure m) s)
% Assm [h4s_lebesgues_lebesgueu_u_monotoneu_u_convergence]: !_1. (!m fi i. happ (happ (happ _1 m) fi) i = h4/lebesgue/pos__fn__integral m (happ fi i)) ==> (!_0. (!fi x i. happ (happ (happ _0 fi) x) i = happ (happ fi i) x) ==> (!m fi f. h4/measure/measure__space m /\ (!i. h4/bool/IN (happ fi i) (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (happ fi i) x)) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) /\ (!x. h4/extreal/ext__mono__increasing (happ (happ _0 fi) x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 fi) x) h4/pred__set/UNIV) = happ f x) ==> h4/lebesgue/pos__fn__integral m f = h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _1 m) fi) h4/pred__set/UNIV)))
% Assm [h4s_lebesgues_lebesgueu_u_monotoneu_u_convergenceu_u_subset]: !_3. (!fi i x. happ (happ (happ _3 fi) i) x = happ (happ fi i) x) ==> (!_2. (!m _1 fi A i. happ (happ (happ (happ (happ _2 m) _1) fi) A) i = h4/lebesgue/pos__fn__integral m (happ (happ _1 (happ (happ _3 fi) i)) A)) ==> (!_1. (!f A x. happ (happ (happ _1 f) A) x = h4/extreal/extreal__mul (happ f x) (happ (h4/measure/indicator__fn A) x)) ==> (!_0. (!fi x i. happ (happ (happ _0 fi) x) i = happ (happ fi i) x) ==> (!m fi f A. h4/measure/measure__space m /\ (!i. h4/bool/IN (happ fi i) (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (happ fi i) x)) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 fi) x) h4/pred__set/UNIV) = happ f x) /\ (!x. h4/extreal/ext__mono__increasing (happ (happ _0 fi) x)) /\ h4/bool/IN A (h4/measure/measurable__sets m) ==> h4/lebesgue/pos__fn__integral m (happ (happ _1 f) A) = h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ (happ (happ _2 m) _1) fi) A) h4/pred__set/UNIV)))))
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_add]: !_0. (!f g x. happ (happ (happ _0 f) g) x = h4/extreal/extreal__add (happ f x) (happ g x)) ==> (!m g f. h4/measure/measure__space m /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x) /\ h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ g x)) /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) /\ h4/bool/IN g (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> h4/lebesgue/pos__fn__integral m (happ (happ _0 f) g) = h4/extreal/extreal__add (h4/lebesgue/pos__fn__integral m f) (h4/lebesgue/pos__fn__integral m g))
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_suminf]: !_2. (!m f i. happ (happ (happ _2 m) f) i = h4/lebesgue/pos__fn__integral m (happ f i)) ==> (!_1. (!f x i. happ (happ (happ _1 f) x) i = happ (happ f i) x) ==> (!_0. (!f x. happ (happ _0 f) x = h4/extreal/ext__suminf (happ (happ _1 f) x)) ==> (!m f. h4/measure/measure__space m /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (happ f i) x)) /\ (!i. h4/bool/IN (happ f i) (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel)) ==> h4/lebesgue/pos__fn__integral m (happ _0 f) = h4/extreal/ext__suminf (happ (happ _2 m) f))))
% Assm [h4s_lebesgues_RADONu_u_Fu_u_def]: !_1. (!f A x. happ (happ (happ _1 f) A) x = h4/extreal/extreal__mul (happ f x) (happ (h4/measure/indicator__fn A) x)) ==> (!_0. (!m v f. ?v. (v <=> h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) /\ (!A. h4/bool/IN A (h4/measure/measurable__sets m) ==> h4/extreal/extreal__le (h4/lebesgue/pos__fn__integral m (happ (happ _1 f) A)) (h4/extreal/Normal (happ (h4/measure/measure v) A)))) /\ happ (happ (happ _0 m) v) f = h4/pair/_2C f v) ==> (!v m. h4/lebesgue/RADON__F m v = h4/pred__set/GSPEC (happ (happ _0 m) v)))
% Assm [h4s_lebesgues_RADONu_u_Fu_u_integralsu_u_def]: !_0. (!m v r. ?v. (v <=> (?f. r = h4/lebesgue/pos__fn__integral m f /\ h4/bool/IN f (h4/lebesgue/RADON__F m v))) /\ happ (happ (happ _0 m) v) r = h4/pair/_2C r v) ==> (!v m. h4/lebesgue/RADON__F__integrals m v = h4/pred__set/GSPEC (happ (happ _0 m) v))
% Assm [h4s_lebesgues_measureu_u_absolutelyu_u_continuousu_u_def]: !v m. h4/lebesgue/measure__absolutely__continuous m v <=> (!A. h4/bool/IN A (h4/measure/measurable__sets m) /\ happ (h4/measure/measure v) A = h4/real/real__of__num h4/num/0 ==> happ (h4/measure/measure m) A = h4/real/real__of__num h4/num/0)
% Assm [h4s_lebesgues_lemmau_u_radonu_u_sequ_u_convu_u_sup]: !_0. (!m f__n n. happ (happ (happ _0 m) f__n) n = h4/lebesgue/pos__fn__integral m (happ f__n n)) ==> (!v m. h4/measure/measure__space m /\ h4/measure/measure__space v /\ h4/measure/m__space v = h4/measure/m__space m /\ h4/measure/measurable__sets v = h4/measure/measurable__sets m ==> (?f__n. (!n. h4/bool/IN (happ f__n n) (h4/lebesgue/RADON__F m v)) /\ (!x n. h4/extreal/extreal__le (happ (happ f__n n) x) (happ (happ f__n (h4/num/SUC n)) x)) /\ h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 m) f__n) h4/pred__set/UNIV) = h4/extreal/extreal__sup (h4/lebesgue/RADON__F__integrals m v)))
% Assm [h4s_lebesgues_RNu_u_lemma2]: !v m. h4/measure/measure__space m /\ h4/measure/measure__space v /\ h4/measure/m__space v = h4/measure/m__space m /\ h4/measure/measurable__sets v = h4/measure/measurable__sets m ==> (?A. h4/bool/IN A (h4/measure/measurable__sets m) /\ h4/real/real__lte (h4/real/real__sub (happ (h4/measure/measure m) (h4/measure/m__space m)) (happ (h4/measure/measure v) (h4/measure/m__space m))) (h4/real/real__sub (happ (h4/measure/measure m) A) (happ (h4/measure/measure v) A)) /\ (!B. h4/bool/IN B (h4/measure/measurable__sets m) /\ h4/pred__set/SUBSET B A ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/real/real__sub (happ (h4/measure/measure m) B) (happ (h4/measure/measure v) B))))
% Goal: !_0. (!f A x. happ (happ (happ _0 f) A) x = h4/extreal/extreal__mul (happ f x) (happ (h4/measure/indicator__fn A) x)) ==> (!v m. h4/measure/measure__space m /\ h4/measure/measure__space v /\ h4/measure/m__space v = h4/measure/m__space m /\ h4/measure/measurable__sets v = h4/measure/measurable__sets m /\ h4/lebesgue/measure__absolutely__continuous v m ==> (?f. h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) /\ (!A. h4/bool/IN A (h4/measure/measurable__sets m) ==> h4/lebesgue/pos__fn__integral m (happ (happ _0 f) A) = h4/extreal/Normal (happ (h4/measure/measure v) A))))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q269922,TV_Q269918]: ![V_f, V_g]: (![V_x]: s(TV_Q269918,happ(s(t_fun(TV_Q269922,TV_Q269918),V_f),s(TV_Q269922,V_x))) = s(TV_Q269918,happ(s(t_fun(TV_Q269922,TV_Q269918),V_g),s(TV_Q269922,V_x))) => s(t_fun(TV_Q269922,TV_Q269918),V_f) = s(t_fun(TV_Q269922,TV_Q269918),V_g))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_bools_ORu_u_DEF, axiom, ![V_x, V_xi_]: (p(s(t_bool,d_or(s(t_bool,V_x),s(t_bool,V_xi_)))) <=> ![V_t]: ((p(s(t_bool,V_x)) => p(s(t_bool,V_t))) => ((p(s(t_bool,V_xi_)) => p(s(t_bool,V_t))) => p(s(t_bool,V_t)))))).
fof(ah4s_bools_Fu_u_DEF, axiom, (p(s(t_bool,f0)) <=> ![V_t]: p(s(t_bool,V_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,f0))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => 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_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f0))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: 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_g),s(TV_u_27a,V_x))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f0) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
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_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) & p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) & (p(s(t_bool,V_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) & p(s(t_bool,V_C))) | p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) | p(s(t_bool,V_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_CONDu_u_RATOR, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = 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_g),s(TV_u_27a,V_x)))))).
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_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_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))))).
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,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
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,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_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_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_markers_Abbrevu_u_def, axiom, ![V_x]: s(t_bool,h4s_markers_abbrev(s(t_bool,V_x))) = s(t_bool,V_x)).
fof(ah4s_combins_Su_u_DEF, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_x0, V_x1]: s(TV_u_27c,h4s_combins_s(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1)))))).
fof(ah4s_combins_Cu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_x, V_x0, V_x1]: s(TV_u_27c,h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27b,V_x0),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,V_x0)))).
fof(ah4s_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_Iu_u_ou_u_IDu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_pairs_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_arithmetics_LESSu_u_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_numerals_numeralu_u_distribu_c17, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_numerals_numeralu_u_distribu_c21, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_equ_c3, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero) <=> p(s(t_bool,f0)))).
fof(ah4s_numerals_numeralu_u_ltu_c1, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))) = s(t_bool,t)).
fof(ah4s_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_predu_u_sets_INu_u_ABS, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x0]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x0))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x0))) => ![V_x, V_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_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_predu_u_sets_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_predu_u_sets_SUBSETu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_INu_u_INTER, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_INTERu_u_SUBSETu_c0, axiom, ![TV_u_27a]: ![V_t, V_s]: p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))),s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ah4s_predu_u_sets_INTERu_u_SUBSETu_c1, axiom, ![TV_u_27a]: ![V_t, V_s]: p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ah4s_predu_u_sets_INu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> ?[V_x]: (s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_countu_u_def, axiom, ![V_uu_0]: (![V_n, V_m]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_bool))),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_bool),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) => ![V_n]: s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_count(s(t_h4s_nums_num,V_n))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_bool))),V_uu_0),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_predu_u_sets_INu_u_COUNT, axiom, ![V_n, V_m]: s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_count(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_predu_u_sets_FINITEu_u_COUNT, axiom, ![V_n]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_count(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_reals_REALu_u_ADDu_u_LID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_reals_REALu_u_MULu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,V_z)))).
fof(ah4s_reals_REALu_u_MULu_u_LINV, axiom, ![V_x]: (~ (s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_realaxs_inv(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))).
fof(ah4s_reals_REALu_u_LTu_u_MUL, axiom, ![V_y, V_x]: ((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_x)))) & 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_y))))) => 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,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))))).
fof(ah4s_reals_realu_u_sub0, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_reals_realu_u_div, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_inv(s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_reals_REALu_u_MULu_u_RID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_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))))))))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_reals_REALu_u_LTu_u_LE, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) <=> (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) & ~ (s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_y))))).
fof(ah4s_reals_REALu_u_LTu_u_IMPu_u_LE, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_reals_REALu_u_LTEu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z)))))).
fof(ah4s_reals_REALu_u_LETu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z)))))).
fof(ah4s_reals_REALu_u_LEu_u_ANTISYM, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x))))) <=> s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_y))).
fof(ah4s_reals_REALu_u_LEu_u_MUL, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_y))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))))).
fof(ah4s_reals_REALu_u_SUBu_u_REFL, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_reals_REALu_u_SUBu_u_0, axiom, ![V_y, V_x]: (s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),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_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_y))).
fof(ah4s_reals_REALu_u_NEGu_u_0, axiom, s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_reals_REALu_u_SUBu_u_LT, axiom, ![V_y, V_x]: 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,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))) = s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_reals_REALu_u_SUBu_u_LE, axiom, ![V_y, V_x]: 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_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))) = s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_reals_REALu_u_LTu_u_IMPu_u_NE, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) => ~ (s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_y)))).
fof(ah4s_reals_REALu_u_LTu_u_HALF1, axiom, ![V_d]: 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,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_d),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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))) = 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_d)))).
fof(ah4s_reals_REALu_u_SUBu_u_RZERO, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_reals_REALu_u_INVu_u_MUL, axiom, ![V_y, V_x]: ((~ (s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) & ~ (s(t_h4s_realaxs_real,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_realaxs_real,h4s_realaxs_inv(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_realaxs_inv(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,h4s_realaxs_inv(s(t_h4s_realaxs_real,V_y))))))).
fof(ah4s_reals_REALu_u_LTu_u_DIV, axiom, ![V_y, V_x]: ((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_x)))) & 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_y))))) => 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,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))))).
fof(ah4s_reals_REALu_u_NEGu_u_HALF, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))) = s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))).
fof(ah4s_reals_equ_u_intsu_c0, axiom, ![V_n, V_m]: (s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,V_m))).
fof(ah4s_reals_ltu_u_intu_c0, axiom, ![V_n, V_m]: 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,V_n))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_seqs_sums0, axiom, ![V_uu_0]: (![V_f, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_reals_sum(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))) => ![V_s, V_f]: 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_s))) = s(t_bool,h4s_seqs_u_2du_2du_3e(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_s))))).
fof(ah4s_seqs_SERu_u_SUB, axiom, ![V_uu_0]: (![V_x, V_y, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y))),s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y),s(t_h4s_nums_num,V_n))))) => ![V_y0, V_y, V_x0, V_x]: ((p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0)))) & p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y),s(t_h4s_realaxs_real,V_y0))))) => p(s(t_bool,h4s_seqs_sums(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_y))),s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x0),s(t_h4s_realaxs_real,V_y0))))))))).
fof(ah4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_COUNT, axiom, ![V_n, V_f]: s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_count(s(t_h4s_nums_num,V_n))))) = s(t_h4s_realaxs_real,h4s_reals_sum(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))),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_utilu_u_probs_SUBSETu_u_INTER2, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) => s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),V_s))).
fof(ah4s_extreals_extrealu_u_ofu_u_numu_u_def, axiom, ![V_n]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = 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,V_n)))))).
fof(ah4s_extreals_extrealu_u_subu_u_defu_c0, axiom, ![V_y, V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sub(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_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_extreals_extrealu_u_leu_u_defu_c0, axiom, ![V_y, V_x]: s(t_bool,h4s_extreals_extrealu_u_le(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_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))).
fof(ah4s_extreals_extrealu_u_leu_u_defu_c0, axiom, ![V_y, V_x]: s(t_bool,h4s_extreals_extrealu_u_le(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_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))).
fof(ah4s_extreals_extrealu_u_leu_u_defu_c1, axiom, ![V_a]: s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,V_a))) = s(t_bool,t)).
fof(ah4s_extreals_extrealu_u_leu_u_defu_c2, axiom, s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,h4s_extreals_posinf))) = s(t_bool,t)).
fof(ah4s_extreals_extrealu_u_leu_u_defu_c3, axiom, ![V_v2]: s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_v2))),s(t_h4s_extreals_extreal,h4s_extreals_posinf))) = s(t_bool,t)).
fof(ah4s_extreals_extrealu_u_leu_u_defu_c4, axiom, s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,h4s_extreals_neginf))) = s(t_bool,f0)).
fof(ah4s_extreals_extrealu_u_leu_u_defu_c5, axiom, ![V_v3]: s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_v3))),s(t_h4s_extreals_extreal,h4s_extreals_neginf))) = s(t_bool,f0)).
fof(ah4s_extreals_extrealu_u_leu_u_defu_c6, axiom, ![V_v5]: s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_v5))))) = s(t_bool,f0)).
fof(ah4s_extreals_extrealu_u_ltu_u_def, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y)))) <=> ~ (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_x))))))).
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_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_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_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_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_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_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_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_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_monou_u_increasingu_u_def, axiom, ![V_f]: (p(s(t_bool,h4s_extreals_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f)))) <=> ![V_m, V_n]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_reals_realu_u_lte(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_m))),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))))))))).
fof(ah4s_extreals_extrealu_u_cases, axiom, ![V_x]: (s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_neginf) | (s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_posinf) | ?[V_r]: s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_r)))))).
fof(ah4s_extreals_extrealu_u_notu_u_inftyu_c0, axiom, ![V_x]: ~ (s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_neginf))).
fof(ah4s_extreals_extrealu_u_notu_u_inftyu_c1, axiom, ![V_x]: ~ (s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf))).
fof(ah4s_extreals_numu_u_notu_u_inftyu_c0, axiom, ![V_n]: ~ (s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,h4s_extreals_neginf))).
fof(ah4s_extreals_extrealu_u_11, axiom, ![V_au_27, V_a]: (s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_a))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_au_27))) <=> s(t_h4s_realaxs_real,V_a) = s(t_h4s_realaxs_real,V_au_27))).
fof(ah4s_extreals_mulu_u_rzero, axiom, ![V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,V_x),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_nums_0)))).
fof(ah4s_extreals_mulu_u_rone, axiom, ![V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,V_x),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,V_x)).
fof(ah4s_extreals_extrealu_u_ltu_u_eq, axiom, ![V_y, V_x]: s(t_bool,h4s_extreals_extrealu_u_lt(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_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))).
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_ltu_u_inftyu_c0, axiom, ![V_y]: p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_y))))))).
fof(ah4s_extreals_ltu_u_inftyu_c1, axiom, ![V_y]: p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_y))),s(t_h4s_extreals_extreal,h4s_extreals_posinf))))).
fof(ah4s_extreals_ltu_u_inftyu_c2, axiom, p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,h4s_extreals_posinf))))).
fof(ah4s_extreals_ltu_u_inftyu_c3, axiom, ![V_x]: ~ (p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,h4s_extreals_neginf)))))).
fof(ah4s_extreals_ltu_u_inftyu_c4, axiom, ![V_x]: ~ (p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,V_x)))))).
fof(ah4s_extreals_ltu_u_inftyu_c5, axiom, ![V_x]: (~ (s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)) <=> p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,h4s_extreals_posinf)))))).
fof(ah4s_extreals_ltu_u_inftyu_c6, axiom, ![V_x]: (~ (s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_neginf)) <=> p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,V_x)))))).
fof(ah4s_extreals_ltu_u_impu_u_le, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y)))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y)))))).
fof(ah4s_extreals_leu_u_trans, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y)))) & p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_z))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_z)))))).
fof(ah4s_extreals_letu_u_trans, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y)))) & p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_z))))) => p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_z)))))).
fof(ah4s_extreals_leu_u_antisym, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y)))) & p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_x))))) <=> s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,V_y))).
fof(ah4s_extreals_lteu_u_trans, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y)))) & p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_z))))) => p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_z)))))).
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_add2, axiom, ![V_z, V_y, V_x, V_w]: ((p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_w),s(t_h4s_extreals_extreal,V_x)))) & p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_z))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_w),s(t_h4s_extreals_extreal,V_y))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_z)))))))).
fof(ah4s_extreals_letu_u_add2, axiom, ![V_z, V_y, V_x, V_w]: ((~ (s(t_h4s_extreals_extreal,V_w) = s(t_h4s_extreals_extreal,h4s_extreals_neginf)) & (~ (s(t_h4s_extreals_extreal,V_w) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)) & (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_w),s(t_h4s_extreals_extreal,V_x)))) & p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_z))))))) => p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_w),s(t_h4s_extreals_extreal,V_y))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_z)))))))).
fof(ah4s_extreals_leu_u_laddu_u_imp, axiom, ![V_z, V_y, V_x]: (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_z)))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_z)))))))).
fof(ah4s_extreals_addu_u_rzero, axiom, ![V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_x),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,V_x)).
fof(ah4s_extreals_subu_u_zerou_u_le, axiom, ![V_y, V_x]: s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y))) = 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_sub(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_x)))))).
fof(ah4s_extreals_subu_u_add2, axiom, ![V_y, V_x]: ((~ (s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_neginf)) & ~ (s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_posinf))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sub(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_x))))) = s(t_h4s_extreals_extreal,V_y))).
fof(ah4s_extreals_extu_u_monou_u_increasingu_u_suc, axiom, ![V_f]: (p(s(t_bool,h4s_extreals_extu_u_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f)))) <=> ![V_n]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))))).
fof(ah4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_MONOu_u_SET, axiom, ![TV_u_27a]: ![V_t, V_s, V_f]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t)))) & (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t)))) => 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_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(TV_u_27a,t_bool),V_t)))))))).
fof(ah4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_NORMAL, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(TV_u_27a,V_x))))) => ![V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_extreals_leu_u_supu_u_imp, axiom, ![V_x, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_extreals_extreal,t_bool),V_p),s(t_h4s_extreals_extreal,V_x)))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),V_p)))))))).
fof(ah4s_extreals_supu_u_le, axiom, ![V_x, V_p]: (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),V_p))),s(t_h4s_extreals_extreal,V_x)))) <=> ![V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_extreals_extreal,t_bool),V_p),s(t_h4s_extreals_extreal,V_y)))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_x))))))).
fof(ah4s_extreals_leu_u_sup, axiom, ![V_x, V_p]: (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),V_p)))))) <=> ![V_y]: (![V_z]: (p(s(t_bool,happ(s(t_fun(t_h4s_extreals_extreal,t_bool),V_p),s(t_h4s_extreals_extreal,V_z)))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_z),s(t_h4s_extreals_extreal,V_y))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y))))))).
fof(ah4s_extreals_supu_u_const, axiom, ![V_uu_0]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_extreals_extreal,t_bool),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_bool)),V_uu_0),s(t_h4s_extreals_extreal,V_x))),s(t_h4s_extreals_extreal,V_y)))) <=> s(t_h4s_extreals_extreal,V_y) = s(t_h4s_extreals_extreal,V_x)) => ![V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_bool)),V_uu_0),s(t_h4s_extreals_extreal,V_x))))) = s(t_h4s_extreals_extreal,V_x))).
fof(ah4s_extreals_supu_u_seq, axiom, ![V_uu_0]: (![V_f, V_n]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(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))))) => ![V_l, V_f]: (p(s(t_bool,h4s_extreals_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f)))) => (p(s(t_bool,h4s_seqs_u_2du_2du_3e(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l)))) <=> s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_l))))))).
fof(ah4s_extreals_extu_u_suminfu_u_def, axiom, ![V_uu_0]: (![V_f, V_n]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f))),s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_count(s(t_h4s_nums_num,V_n))))) => ![V_f]: s(t_h4s_extreals_extreal,h4s_extreals_extu_u_suminf(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))))).
fof(ah4s_extreals_extu_u_suminfu_u_cmul, axiom, ![V_uu_0]: (![V_c, V_f, V_n]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_h4s_extreals_extreal,V_c))),s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f))),s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,V_c),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))))) => ![V_f, V_c]: ((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,V_c)))) & ![V_n]: 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(t_h4s_nums_num,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))))))) => s(t_h4s_extreals_extreal,h4s_extreals_extu_u_suminf(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_h4s_extreals_extreal,V_c))),s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,V_c),s(t_h4s_extreals_extreal,h4s_extreals_extu_u_suminf(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_f)))))))).
fof(ah4s_measures_spaceu_u_def, axiom, ![TV_u_27a]: ![V_y, V_x]: s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y))))) = s(t_fun(TV_u_27a,t_bool),V_x)).
fof(ah4s_measures_subsetsu_u_def, axiom, ![TV_u_27a]: ![V_y, V_x]: s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y)).
fof(ah4s_measures_mu_u_spaceu_u_def, axiom, ![TV_u_27a]: ![V_sts, V_sp, V_mu]: s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_mu))))))) = s(t_fun(TV_u_27a,t_bool),V_sp)).
fof(ah4s_measures_measurableu_u_setsu_u_def, axiom, ![TV_u_27a]: ![V_sts, V_sp, V_mu]: s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_mu))))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts)).
fof(ah4s_measures_measureu_u_def, axiom, ![TV_u_27a]: ![V_sts, V_sp, V_mu]: s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_mu))))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_mu)).
fof(ah4s_measures_positiveu_u_def, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) <=> (s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & ![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_s)))))))))).
fof(ah4s_measures_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),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),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),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),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_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),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (p(s(t_bool,h4s_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_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_measures_INCREASING, axiom, ![TV_u_27a]: ![V_t, V_s, 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)))) & (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_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_t)))))))).
fof(ah4s_measures_MEASUREu_u_EMPTY, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_measures_MEASUREu_u_SPACEu_u_INCREASING, 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_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_MEASUREu_u_SPACEu_u_POSITIVE, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) => p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))).
fof(ah4s_measures_MEASUREu_u_SPACEu_u_INTER, axiom, ![TV_u_27a]: ![V_t, V_s, V_m]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))).
fof(ah4s_measures_MEASUREu_u_SPACEu_u_SUBSETu_u_MSPACE, axiom, ![TV_u_27a]: ![V_m, V_A]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_A),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))) => p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_A),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))).
fof(ah4s_measures_MEASUREu_u_SPACEu_u_EMPTYu_u_MEASURABLE, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))).
fof(ah4s_measures_MEASUREu_u_SPACEu_u_MSPACEu_u_MEASURABLE, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))).
fof(ah4s_measures_MEASUREu_u_SPACEu_u_CMUL, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_c, V_m, 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)),happ(s(t_fun(t_h4s_realaxs_real,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))),V_uu_0),s(t_h4s_realaxs_real,V_c))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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))) = 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),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))))) => ![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_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))))) => p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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)),happ(s(t_fun(t_h4s_realaxs_real,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))),V_uu_0),s(t_h4s_realaxs_real,V_c))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_INu_u_MEASURABLEu_u_BORELu_u_INDICATOR, axiom, ![TV_u_27a]: ![V_f, V_a, 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_bools_in(s(t_fun(TV_u_27a,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)))))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_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_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,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_A))),s(TV_u_27a,V_x)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),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)),V_a),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_measures_INu_u_MEASURABLEu_u_BORELu_u_CMUL, axiom, ![TV_u_27a]: ![V_z, V_g, V_f, 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_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),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)),V_a),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)))))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_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_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),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_z))),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_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),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)),V_a),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_measures_INu_u_MEASURABLEu_u_BORELu_u_ADD, axiom, ![TV_u_27a]: ![V_h, V_g, V_f, 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_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),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)),V_a),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)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),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)),V_a),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)))))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_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_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_h),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x))))))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_h),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)),V_a),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_measures_INu_u_MEASURABLEu_u_BORELu_u_MULu_u_INDICATOR, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_f, 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_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_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,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,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))))) => ![V_s, V_f, 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_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),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)),V_a),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)))))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_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_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))),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)),V_a),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_measures_INu_u_MEASURABLEu_u_BORELu_u_MONOu_u_SUP, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_fn, V_x, V_n]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fn))),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fn),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) => ![V_fn, V_f, 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)))) & (![V_n]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fn),s(t_h4s_nums_num,V_n))),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)),V_a),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)))))) & (![V_n, V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_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)))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fn),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fn),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))),s(TV_u_27a,V_x))))))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_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_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fn))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),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)),V_a),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_measures_indicatoru_u_fnu_u_suminf, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_a, V_x, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a))),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_i))) = 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),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_h4s_nums_num,V_i))))),s(TV_u_27a,V_x))) => ![V_x, V_a]: (![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_a),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_a),s(t_h4s_nums_num,V_n))))))) => s(t_h4s_extreals_extreal,h4s_extreals_extu_u_suminf(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a))),s(TV_u_27a,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),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_a),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))))),s(TV_u_27a,V_x)))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_mspace, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_m, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_indicatoru_u_fn(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(TV_u_27a,V_x))))) => ![V_m, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & ![V_x]: 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))))))) => 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_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_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_zero, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_uu_0),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) => ![V_m]: (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) => s(t_h4s_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_uu_0))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_pos, axiom, ![TV_u_27a]: ![V_m, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & ![V_x]: 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_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_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_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),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_lebesgues_posu_u_fnu_u_integralu_u_indicator, axiom, ![TV_u_27a]: ![V_s, V_m]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_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),h4s_measures_indicatoru_u_fn(s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),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_lebesgues_lebesgueu_u_monotoneu_u_convergence, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_m, V_fi, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_h4s_extreals_extreal)),V_fi))),s(t_h4s_nums_num,V_i))) = 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_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))))) => ![V_uu_0]: (![V_fi, V_x, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))) => ![V_m, V_fi, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (![V_i]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),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),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),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)))))) & (![V_i, V_x]: 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),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x)))))) & (![V_x]: 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)))))) & (![V_x]: p(s(t_bool,h4s_extreals_extu_u_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x)))))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))))))))) => s(t_h4s_extreals_extreal,h4s_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_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_h4s_extreals_extreal)),V_fi))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))))))).
fof(ah4s_lebesgues_lebesgueu_u_monotoneu_u_convergenceu_u_subset, axiom, ![TV_u_27a]: ![V_uu_3]: (![V_fi, V_i, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_3),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))) => ![V_uu_2]: (![V_m, V_uu_1, V_fi, V_A, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))))),V_uu_2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_1))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(t_fun(TV_u_27a,t_bool),V_A))),s(t_h4s_nums_num,V_i))) = 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_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_3),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(t_h4s_nums_num,V_i))))),s(t_fun(TV_u_27a,t_bool),V_A))))) => ![V_uu_1]: (![V_f, V_A, 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_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_bool),V_A))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,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_A))),s(TV_u_27a,V_x))))) => ![V_uu_0]: (![V_fi, V_x, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))) => ![V_m, V_fi, 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)))) & (![V_i]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),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),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),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)))))) & (![V_i, V_x]: 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),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x)))))) & (![V_x]: 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)))))) & (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))) = 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_x]: p(s(t_bool,h4s_extreals_extu_u_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x)))))) & 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),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_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_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_bool),V_A))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))))),V_uu_2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_1))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(t_fun(TV_u_27a,t_bool),V_A))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))))))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_add, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_g, 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_fun(TV_u_27a,t_h4s_extreals_extreal),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_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x))))) => ![V_m, V_g, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (![V_x]: (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_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_g),s(TV_u_27a,V_x))))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),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),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),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)))))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),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),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel))))))))) => s(t_h4s_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_fun(TV_u_27a,t_h4s_extreals_extreal),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_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(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_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_g)))))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_suminf, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_m, V_f, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_h4s_extreals_extreal)),V_f))),s(t_h4s_nums_num,V_i))) = 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_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(t_h4s_nums_num,V_i))))) => ![V_uu_1]: (![V_f, V_x, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))) => ![V_uu_0]: (![V_f, V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extu_u_suminf(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))),s(TV_u_27a,V_x))))) => ![V_m, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (![V_i, V_x]: 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),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x)))))) & ![V_i]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f),s(t_h4s_nums_num,V_i))),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),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))) => s(t_h4s_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(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_f))))) = s(t_h4s_extreals_extreal,h4s_extreals_extu_u_suminf(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_h4s_extreals_extreal)),V_f)))))))))).
fof(ah4s_lebesgues_RADONu_u_Fu_u_def, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_f, V_A, 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_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_bool),V_A))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,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_A))),s(TV_u_27a,V_x))))) => ![V_uu_0]: (![V_m, V_v, V_f]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),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),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),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)))))) & (![V_x]: 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)))))) & ![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),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_extreals_extrealu_u_le(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_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_bool),V_A))))),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))),s(t_fun(TV_u_27a,t_bool),V_A)))))))))))) & s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_extreals_extreal),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_h4s_extreals_extreal),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_extreals_extreal),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_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool)))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))) = s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_bool,V_v0)))) => ![V_v, V_m]: s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_lebesgues_radonu_u_f(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))) = s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_extreals_extreal),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_h4s_extreals_extreal),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_extreals_extreal),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_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool)))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v)))))))).
fof(ah4s_lebesgues_RADONu_u_Fu_u_integralsu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_m, V_v, V_r]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> ?[V_f]: (s(t_h4s_extreals_extreal,V_r) = 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))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_lebesgues_radonu_u_f(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v)))))))) & s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool),happ(s(t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,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_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,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_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(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_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool)))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))),s(t_h4s_extreals_extreal,V_r))) = s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool),h4s_pairs_u_2c(s(t_h4s_extreals_extreal,V_r),s(t_bool,V_v0)))) => ![V_v, V_m]: s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_radonu_u_fu_u_integrals(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))) = s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,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_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,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_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(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_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool)))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))))))).
fof(ah4s_lebesgues_measureu_u_absolutelyu_u_continuousu_u_def, axiom, ![TV_u_27a]: ![V_v, V_m]: (p(s(t_bool,h4s_lebesgues_measureu_u_absolutelyu_u_continuous(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v)))) <=> ![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),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))),s(t_fun(TV_u_27a,t_bool),V_A))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_A))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_lebesgues_lemmau_u_radonu_u_sequ_u_convu_u_sup, axiom, ![TV_u_27b]: ![V_uu_0]: (![V_m, V_fu_u_n, V_n]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_extreals_extreal)),V_fu_u_n))),s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(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_m),s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_extreals_extreal)),V_fu_u_n),s(t_h4s_nums_num,V_n))))) => ![V_v, V_m]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_v)))) & (s(t_fun(TV_u_27b,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_v))) = s(t_fun(TV_u_27b,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m))) & s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_v))) = s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m)))))) => ?[V_fu_u_n]: (![V_n]: p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_extreals_extreal)),V_fu_u_n),s(t_h4s_nums_num,V_n))),s(t_fun(t_fun(TV_u_27b,t_h4s_extreals_extreal),t_bool),h4s_lebesgues_radonu_u_f(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_m),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_v)))))) & (![V_x, V_n]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_extreals_extreal)),V_fu_u_n),s(t_h4s_nums_num,V_n))),s(TV_u_27b,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27b,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_extreals_extreal)),V_fu_u_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))),s(TV_u_27b,V_x)))))) & s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27b,t_bool),t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_extreals_extreal)),V_fu_u_n))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_radonu_u_fu_u_integrals(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_m),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_v)))))))))).
fof(ah4s_lebesgues_RNu_u_lemma2, axiom, ![TV_u_27a]: ![V_v, 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_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_v)))) & (s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))) = s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))) & s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => ?[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),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_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))),s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_A))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))),s(t_fun(TV_u_27a,t_bool),V_A)))))))) & ![V_B]: ((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),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_B),s(t_fun(TV_u_27a,t_bool),V_A))))) => 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_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_B))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))),s(t_fun(TV_u_27a,t_bool),V_B))))))))))))).
fof(ch4s_lebesgues_Radonu_u_Nikodym, conjecture, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_A, 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_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_bool),V_A))),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,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_A))),s(TV_u_27a,V_x))))) => ![V_v, 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_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_v)))) & (s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))) = s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))) & (s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))) & p(s(t_bool,h4s_lebesgues_measureu_u_absolutelyu_u_continuous(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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(TV_u_27a,t_h4s_extreals_extreal),V_f),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),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),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)))))) & (![V_x]: 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)))))) & ![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),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_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_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(TV_u_27a,t_bool),V_A))))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_v))),s(t_fun(TV_u_27a,t_bool),V_A))))))))))).
