%   ORIGINAL: h4/lebesgue/lemma__fn__sup
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_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/SELECT__ELIM__THM: !Q P. (?x. P x) /\ (!x. P x ==> Q x) ==> Q (h4/min/_40 P)
% 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/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% 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/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/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/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/arithmetic/LESS__EQ__ADD: !n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/LESS__EQ__REFL: !m. h4/arithmetic/_3C_3D m m
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/GREATER__EQ: !n m. h4/arithmetic/_3E_3D n m <=> h4/arithmetic/_3C_3D m n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/numeral/numeral__distrib_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__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c28: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/numeral/numeral__lt_c1: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT2 n) <=> T
% Assm: h4/numeral/numeral__lte_c0: !n. h4/arithmetic/_3C_3D h4/arithmetic/ZERO n <=> T
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__lte_c4: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT2 m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% 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/real/REAL__LT__TRANS: !z y x. h4/realax/real__lt x y /\ h4/realax/real__lt 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__LT__IMP__NE: !y x. h4/realax/real__lt x y ==> ~(x = y)
% Assm: h4/real/REAL__LT__ADDR: !y x. h4/realax/real__lt x (h4/realax/real__add x y) <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) y
% Assm: h4/real/REAL__LT__SUB__RADD: !z y x. h4/realax/real__lt (h4/real/real__sub x y) z <=> h4/realax/real__lt x (h4/realax/real__add z y)
% Assm: h4/real/ABS__BETWEEN: !y x d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) d /\ h4/realax/real__lt (h4/real/real__sub x d) y /\ h4/realax/real__lt y (h4/realax/real__add x d) <=> h4/realax/real__lt (h4/real/abs (h4/real/real__sub y x)) d
% Assm: h4/real/POW__ONE: !n. h4/real/pow (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) n = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/real/REAL__POW__LT: !x n. 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) (h4/real/pow x n)
% Assm: h4/real/REAL__POW__DIV: !y x n. h4/real/pow (h4/real/_2F x y) n = h4/real/_2F (h4/real/pow x n) (h4/real/pow y n)
% Assm: h4/real/REAL__DIV__ADD: !z y x. h4/realax/real__add (h4/real/_2F y x) (h4/real/_2F z x) = h4/real/_2F (h4/realax/real__add y z) x
% Assm: h4/real/le__int_c0: !n m. h4/real/real__lte (h4/real/real__of__num n) (h4/real/real__of__num m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/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/SEQ: !x0 x. h4/seq/_2D_2D_3E x x0 <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?N. !n. h4/arithmetic/_3E_3D n N ==> h4/realax/real__lt (h4/real/abs (h4/real/real__sub (x n) x0)) e))
% Assm: h4/util__prob/POW__HALF__POS: !n. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/pow (h4/real/_2F (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) n)
% Assm: h4/util__prob/POW__HALF__SMALL: !e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?n. h4/realax/real__lt (h4/real/pow (h4/real/_2F (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) n) e)
% Assm: h4/util__prob/POW__HALF__MONO: !n m. h4/arithmetic/_3C_3D m n ==> h4/real/real__lte (h4/real/pow (h4/real/_2F (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) n) (h4/real/pow (h4/real/_2F (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) m)
% 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__add__def_c0: !y x. h4/extreal/extreal__add (h4/extreal/Normal x) (h4/extreal/Normal y) = h4/extreal/Normal (h4/realax/real__add x y)
% 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__pow__def_c0: !n a. h4/extreal/extreal__pow (h4/extreal/Normal a) n = h4/extreal/Normal (h4/real/pow a n)
% Assm: h4/extreal/extreal__pow__def_c0: !n a. h4/extreal/extreal__pow (h4/extreal/Normal a) n = h4/extreal/Normal (h4/real/pow a n)
% Assm: h4/extreal/extreal__pow__def_c1: !n. h4/extreal/extreal__pow h4/extreal/PosInf n = h4/bool/COND (n = h4/num/0) (h4/extreal/Normal (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) h4/extreal/PosInf
% Assm: h4/extreal/extreal__pow__def_c2: !n. h4/extreal/extreal__pow h4/extreal/NegInf n = h4/bool/COND (n = h4/num/0) (h4/extreal/Normal (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/bool/COND (h4/arithmetic/EVEN n) h4/extreal/PosInf h4/extreal/NegInf)
% 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/extreal__11: !a_27 a. h4/extreal/Normal a = h4/extreal/Normal a_27 <=> a = a_27
% 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__infty_c0: !x. h4/extreal/extreal__le h4/extreal/NegInf x
% Assm: h4/extreal/le__infty_c1: !x. h4/extreal/extreal__le x h4/extreal/PosInf
% Assm: h4/extreal/le__infty_c1: !x. h4/extreal/extreal__le x h4/extreal/PosInf
% Assm: h4/extreal/le__infty_c2: !x. h4/extreal/extreal__le x h4/extreal/NegInf <=> x = h4/extreal/NegInf
% Assm: h4/extreal/le__infty_c3: !x. h4/extreal/extreal__le h4/extreal/PosInf x <=> x = h4/extreal/PosInf
% Assm: h4/extreal/le__infty_c3: !x. h4/extreal/extreal__le h4/extreal/PosInf x <=> x = h4/extreal/PosInf
% 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/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/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/extreal__div__eq: !y x. h4/extreal/extreal__div (h4/extreal/Normal x) (h4/extreal/Normal y) = h4/extreal/Normal (h4/real/_2F x y)
% Assm: h4/extreal/pow__le__mono: !x n m. h4/extreal/extreal__le (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) x /\ h4/arithmetic/_3C_3D n m ==> h4/extreal/extreal__le (h4/extreal/extreal__pow x n) (h4/extreal/extreal__pow x m)
% Assm: h4/extreal/ext__mono__increasing__def: !f. h4/extreal/ext__mono__increasing f <=> (!m n. h4/arithmetic/_3C_3D m n ==> h4/extreal/extreal__le (f m) (f n))
% Assm: h4/extreal/EXTREAL__ARCH__POW: !x. ~(x = h4/extreal/PosInf) ==> (?n. h4/extreal/extreal__lt x (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n))
% Assm: h4/extreal/sup__eq: !x p. h4/extreal/extreal__sup p = x <=> (!y. p y ==> h4/extreal/extreal__le y x) /\ (!y. (!z. p z ==> h4/extreal/extreal__le z y) ==> h4/extreal/extreal__le x y)
% 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/lebesgue/lemma__fn__1: !x n m k f. h4/bool/IN x (h4/measure/m__space m) /\ h4/bool/IN k (h4/pred__set/count (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) n)) /\ h4/extreal/extreal__le (h4/extreal/extreal__div (h4/extreal/extreal__of__num k) (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n)) (f x) /\ h4/extreal/extreal__lt (f x) (h4/extreal/extreal__div (h4/extreal/extreal__add (h4/extreal/extreal__of__num k) (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n)) ==> h4/lebesgue/fn__seq m f n x = h4/extreal/extreal__div (h4/extreal/extreal__of__num k) (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n)
% Assm: h4/lebesgue/lemma__fn__2: !x n m f. h4/bool/IN x (h4/measure/m__space m) /\ h4/extreal/extreal__le (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n) (f x) ==> h4/lebesgue/fn__seq m f n x = h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n
% Assm: h4/lebesgue/lemma__fn__3: !x n m f. h4/bool/IN x (h4/measure/m__space m) /\ h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x) ==> h4/extreal/extreal__le (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n) (f x) \/ (?k. h4/bool/IN k (h4/pred__set/count (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) n)) /\ h4/extreal/extreal__le (h4/extreal/extreal__div (h4/extreal/extreal__of__num k) (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n)) (f x) /\ h4/extreal/extreal__lt (f x) (h4/extreal/extreal__div (h4/extreal/extreal__add (h4/extreal/extreal__of__num k) (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n)))
% Assm: h4/lebesgue/lemma__fn__5: !x n m f. 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/fn__seq m f n x)
% Assm: h4/lebesgue/lemma__fn__mono__increasing: !x m f. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x) ==> h4/extreal/ext__mono__increasing (\n. h4/lebesgue/fn__seq m f n x)
% Assm: h4/lebesgue/lemma__fn__upper__bounded: !x n m f. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x) ==> h4/extreal/extreal__le (h4/lebesgue/fn__seq m f n x) (f x)
% Goal: !x m f. h4/bool/IN x (h4/measure/m__space m) /\ h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (\n. h4/lebesgue/fn__seq m f n x) h4/pred__set/UNIV) = f x
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_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_SELECTu_u_ELIMu_u_THM]: !Q P. (?x. happ P x) /\ (!x. happ P x ==> happ Q x) ==> happ Q (h4/min/_40 P)
% 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_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% 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_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_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_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_arithmetics_LESSu_u_EQu_u_ADD]: !n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_LESSu_u_EQu_u_REFL]: !m. h4/arithmetic/_3C_3D m m
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_GREATERu_u_EQ]: !n m. h4/arithmetic/_3E_3D n m <=> h4/arithmetic/_3C_3D m n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_numerals_numeralu_u_distribu_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_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c28]: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_numerals_numeralu_u_ltu_c1]: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT2 n) <=> T
% Assm [h4s_numerals_numeralu_u_lteu_c0]: !n. h4/arithmetic/_3C_3D h4/arithmetic/ZERO n <=> T
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_lteu_c4]: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT2 m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% 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_reals_REALu_u_LTu_u_TRANS]: !z y x. h4/realax/real__lt x y /\ h4/realax/real__lt 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_LTu_u_IMPu_u_NE]: !y x. h4/realax/real__lt x y ==> ~(x = y)
% Assm [h4s_reals_REALu_u_LTu_u_ADDR]: !y x. h4/realax/real__lt x (h4/realax/real__add x y) <=> h4/realax/real__lt (h4/real/real__of__num h4/num/0) y
% Assm [h4s_reals_REALu_u_LTu_u_SUBu_u_RADD]: !z y x. h4/realax/real__lt (h4/real/real__sub x y) z <=> h4/realax/real__lt x (h4/realax/real__add z y)
% Assm [h4s_reals_ABSu_u_BETWEEN]: !y x d. h4/realax/real__lt (h4/real/real__of__num h4/num/0) d /\ h4/realax/real__lt (h4/real/real__sub x d) y /\ h4/realax/real__lt y (h4/realax/real__add x d) <=> h4/realax/real__lt (h4/real/abs (h4/real/real__sub y x)) d
% Assm [h4s_reals_POWu_u_ONE]: !n. h4/real/pow (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) n = h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_reals_REALu_u_POWu_u_LT]: !x n. 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) (h4/real/pow x n)
% Assm [h4s_reals_REALu_u_POWu_u_DIV]: !y x n. h4/real/pow (h4/real/_2F x y) n = h4/real/_2F (h4/real/pow x n) (h4/real/pow y n)
% Assm [h4s_reals_REALu_u_DIVu_u_ADD]: !z y x. h4/realax/real__add (h4/real/_2F y x) (h4/real/_2F z x) = h4/real/_2F (h4/realax/real__add y z) x
% Assm [h4s_reals_leu_u_intu_c0]: !n m. h4/real/real__lte (h4/real/real__of__num n) (h4/real/real__of__num m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_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_SEQ]: !x0 x. h4/seq/_2D_2D_3E x x0 <=> (!e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?N. !n. h4/arithmetic/_3E_3D n N ==> h4/realax/real__lt (h4/real/abs (h4/real/real__sub (happ x n) x0)) e))
% Assm [h4s_utilu_u_probs_POWu_u_HALFu_u_POS]: !n. h4/realax/real__lt (h4/real/real__of__num h4/num/0) (h4/real/pow (h4/real/_2F (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) n)
% Assm [h4s_utilu_u_probs_POWu_u_HALFu_u_SMALL]: !e. h4/realax/real__lt (h4/real/real__of__num h4/num/0) e ==> (?n. h4/realax/real__lt (h4/real/pow (h4/real/_2F (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) n) e)
% Assm [h4s_utilu_u_probs_POWu_u_HALFu_u_MONO]: !n m. h4/arithmetic/_3C_3D m n ==> h4/real/real__lte (h4/real/pow (h4/real/_2F (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) n) (h4/real/pow (h4/real/_2F (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) m)
% 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_addu_u_defu_c0]: !y x. h4/extreal/extreal__add (h4/extreal/Normal x) (h4/extreal/Normal y) = h4/extreal/Normal (h4/realax/real__add x y)
% 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_powu_u_defu_c0]: !n a. h4/extreal/extreal__pow (h4/extreal/Normal a) n = h4/extreal/Normal (h4/real/pow a n)
% Assm [h4s_extreals_extrealu_u_powu_u_defu_c0]: !n a. h4/extreal/extreal__pow (h4/extreal/Normal a) n = h4/extreal/Normal (h4/real/pow a n)
% Assm [h4s_extreals_extrealu_u_powu_u_defu_c1]: !n. ?v. (v <=> n = h4/num/0) /\ h4/extreal/extreal__pow h4/extreal/PosInf n = h4/bool/COND v (h4/extreal/Normal (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) h4/extreal/PosInf
% Assm [h4s_extreals_extrealu_u_powu_u_defu_c2]: !n. ?v. (v <=> n = h4/num/0) /\ h4/extreal/extreal__pow h4/extreal/NegInf n = h4/bool/COND v (h4/extreal/Normal (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/bool/COND (h4/arithmetic/EVEN n) h4/extreal/PosInf h4/extreal/NegInf)
% 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_extrealu_u_11]: !a_27 a. h4/extreal/Normal a = h4/extreal/Normal a_27 <=> a = a_27
% 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_inftyu_c0]: !x. h4/extreal/extreal__le h4/extreal/NegInf x
% Assm [h4s_extreals_leu_u_inftyu_c1]: !x. h4/extreal/extreal__le x h4/extreal/PosInf
% Assm [h4s_extreals_leu_u_inftyu_c1]: !x. h4/extreal/extreal__le x h4/extreal/PosInf
% Assm [h4s_extreals_leu_u_inftyu_c2]: !x. h4/extreal/extreal__le x h4/extreal/NegInf <=> x = h4/extreal/NegInf
% Assm [h4s_extreals_leu_u_inftyu_c3]: !x. h4/extreal/extreal__le h4/extreal/PosInf x <=> x = h4/extreal/PosInf
% Assm [h4s_extreals_leu_u_inftyu_c3]: !x. h4/extreal/extreal__le h4/extreal/PosInf x <=> x = h4/extreal/PosInf
% 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_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_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_extrealu_u_divu_u_eq]: !y x. h4/extreal/extreal__div (h4/extreal/Normal x) (h4/extreal/Normal y) = h4/extreal/Normal (h4/real/_2F x y)
% Assm [h4s_extreals_powu_u_leu_u_mono]: !x n m. h4/extreal/extreal__le (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) x /\ h4/arithmetic/_3C_3D n m ==> h4/extreal/extreal__le (h4/extreal/extreal__pow x n) (h4/extreal/extreal__pow x m)
% Assm [h4s_extreals_extu_u_monou_u_increasingu_u_def]: !f. h4/extreal/ext__mono__increasing f <=> (!m n. h4/arithmetic/_3C_3D m n ==> h4/extreal/extreal__le (happ f m) (happ f n))
% Assm [h4s_extreals_EXTREALu_u_ARCHu_u_POW]: !x. ~(x = h4/extreal/PosInf) ==> (?n. h4/extreal/extreal__lt x (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n))
% Assm [h4s_extreals_supu_u_eq]: !x p. h4/extreal/extreal__sup p = x <=> (!y. happ p y ==> h4/extreal/extreal__le y x) /\ (!y. (!z. happ p z ==> h4/extreal/extreal__le z y) ==> h4/extreal/extreal__le x y)
% 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_lebesgues_lemmau_u_fnu_u_1]: !x n m k f. h4/bool/IN x (h4/measure/m__space m) /\ h4/bool/IN k (h4/pred__set/count (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) n)) /\ h4/extreal/extreal__le (h4/extreal/extreal__div (h4/extreal/extreal__of__num k) (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n)) (happ f x) /\ h4/extreal/extreal__lt (happ f x) (h4/extreal/extreal__div (h4/extreal/extreal__add (h4/extreal/extreal__of__num k) (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n)) ==> h4/lebesgue/fn__seq m f n x = h4/extreal/extreal__div (h4/extreal/extreal__of__num k) (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n)
% Assm [h4s_lebesgues_lemmau_u_fnu_u_2]: !x n m f. h4/bool/IN x (h4/measure/m__space m) /\ h4/extreal/extreal__le (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n) (happ f x) ==> h4/lebesgue/fn__seq m f n x = h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n
% Assm [h4s_lebesgues_lemmau_u_fnu_u_3]: !x n m f. 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/extreal/extreal__le (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n) (happ f x) \/ (?k. h4/bool/IN k (h4/pred__set/count (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) n)) /\ h4/extreal/extreal__le (h4/extreal/extreal__div (h4/extreal/extreal__of__num k) (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n)) (happ f x) /\ h4/extreal/extreal__lt (happ f x) (h4/extreal/extreal__div (h4/extreal/extreal__add (h4/extreal/extreal__of__num k) (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/extreal/extreal__pow (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) n)))
% Assm [h4s_lebesgues_lemmau_u_fnu_u_5]: !x n m f. 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/fn__seq m f n x)
% Assm [h4s_lebesgues_lemmau_u_fnu_u_monou_u_increasing]: !_0. (!m f x n. happ (happ (happ (happ _0 m) f) x) n = h4/lebesgue/fn__seq m f n x) ==> (!x m f. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x) ==> h4/extreal/ext__mono__increasing (happ (happ (happ _0 m) f) x))
% Assm [h4s_lebesgues_lemmau_u_fnu_u_upperu_u_bounded]: !x n m f. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x) ==> h4/extreal/extreal__le (h4/lebesgue/fn__seq m f n x) (happ f x)
% Goal: !_0. (!m f x n. happ (happ (happ (happ _0 m) f) x) n = h4/lebesgue/fn__seq m f n x) ==> (!x m f. 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/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ (happ _0 m) f) x) h4/pred__set/UNIV) = happ f x)
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_Q268248,TV_Q268244]: ![V_f, V_g]: (![V_x]: s(TV_Q268244,happ(s(t_fun(TV_Q268248,TV_Q268244),V_f),s(TV_Q268248,V_x))) = s(TV_Q268244,happ(s(t_fun(TV_Q268248,TV_Q268244),V_g),s(TV_Q268248,V_x))) => s(t_fun(TV_Q268248,TV_Q268244),V_f) = s(t_fun(TV_Q268248,TV_Q268244),V_g))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,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_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_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) | 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_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_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_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_SELECTu_u_ELIMu_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)))) & ![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)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
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_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) & p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) & (p(s(t_bool,V_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) & p(s(t_bool,V_C))) | p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) | p(s(t_bool,V_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> s(t_bool,V_t) = s(t_bool,f0))).
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_EQu_u_EXPAND, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) | (~ (p(s(t_bool,V_t1))) & ~ (p(s(t_bool,V_t2))))))).
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_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_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_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_ADD, axiom, ![V_n, V_m]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_REFL, axiom, ![V_m]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_m))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_GREATERu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3eu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_NOTu_u_LEQ, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_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_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_distribu_c28, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_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_numerals_numeralu_u_lteu_c0, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))) = s(t_bool,t)).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f0)).
fof(ah4s_numerals_numeralu_u_lteu_c4, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_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_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_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_reals_REALu_u_LTu_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_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_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_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_ADDR, axiom, ![V_y, V_x]: s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))) = s(t_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)))).
fof(ah4s_reals_REALu_u_LTu_u_SUBu_u_RADD, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_realaxs_realu_u_lt(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,V_z))) = s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_z),s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_reals_ABSu_u_BETWEEN, axiom, ![V_y, V_x, V_d]: ((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_d)))) & (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_d))),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,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_d)))))))) <=> p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x))))),s(t_h4s_realaxs_real,V_d)))))).
fof(ah4s_reals_POWu_u_ONE, axiom, ![V_n]: s(t_h4s_realaxs_real,h4s_reals_pow(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_nums_num,V_n))) = 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_POWu_u_LT, axiom, ![V_x, V_n]: (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,h4s_reals_pow(s(t_h4s_realaxs_real,V_x),s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_reals_REALu_u_POWu_u_DIV, axiom, ![V_y, V_x, V_n]: s(t_h4s_realaxs_real,h4s_reals_pow(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_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_x),s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_y),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_reals_REALu_u_DIVu_u_ADD, axiom, ![V_z, V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_z),s(t_h4s_realaxs_real,V_x))))) = s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_reals_leu_u_intu_c0, axiom, ![V_n, V_m]: s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_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_SEQ, axiom, ![V_x0, V_x]: (p(s(t_bool,h4s_seqs_u_2du_2du_3e(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_realaxs_real,V_x0)))) <=> ![V_e]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_e)))) => ?[V_N]: ![V_n]: (p(s(t_bool,h4s_arithmetics_u_3eu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_N)))) => p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_abs(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,V_x0))))),s(t_h4s_realaxs_real,V_e)))))))).
fof(ah4s_utilu_u_probs_POWu_u_HALFu_u_POS, axiom, ![V_n]: 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_pow(s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),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_nums_num,V_n))))))).
fof(ah4s_utilu_u_probs_POWu_u_HALFu_u_SMALL, axiom, ![V_e]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_e)))) => ?[V_n]: p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),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_nums_num,V_n))),s(t_h4s_realaxs_real,V_e)))))).
fof(ah4s_utilu_u_probs_POWu_u_HALFu_u_MONO, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),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_nums_num,V_n))),s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),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_nums_num,V_m)))))))).
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_addu_u_defu_c0, axiom, ![V_y, V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(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_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))).
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_powu_u_defu_c0, axiom, ![V_n, V_a]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_pow(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_a))),s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_a),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_extreals_extrealu_u_powu_u_defu_c0, axiom, ![V_n, V_a]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_pow(s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_a))),s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_a),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_extreals_extrealu_u_powu_u_defu_c1, axiom, ![V_n]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_pow(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_nums_num,V_n))) = 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_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_posinf))))).
fof(ah4s_extreals_extrealu_u_powu_u_defu_c2, axiom, ![V_n]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_pow(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_nums_num,V_n))) = 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_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))),s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,h4s_extreals_neginf))))))).
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_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_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_inftyu_c0, axiom, ![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,V_x))))).
fof(ah4s_extreals_leu_u_inftyu_c1, axiom, ![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_posinf))))).
fof(ah4s_extreals_leu_u_inftyu_c1, axiom, ![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_posinf))))).
fof(ah4s_extreals_leu_u_inftyu_c2, axiom, ![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_neginf)))) <=> s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_neginf))).
fof(ah4s_extreals_leu_u_inftyu_c3, axiom, ![V_x]: (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,V_x)))) <=> s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_posinf))).
fof(ah4s_extreals_leu_u_inftyu_c3, axiom, ![V_x]: (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,V_x)))) <=> s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_posinf))).
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_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_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_extrealu_u_divu_u_eq, axiom, ![V_y, V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_div(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_u_2f(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_extreals_powu_u_leu_u_mono, axiom, ![V_x, V_n, 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_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)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_pow(s(t_h4s_extreals_extreal,V_x),s(t_h4s_nums_num,V_n))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_pow(s(t_h4s_extreals_extreal,V_x),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_extreals_extu_u_monou_u_increasingu_u_def, 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_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_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_m))),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))))))))).
fof(ah4s_extreals_EXTREALu_u_ARCHu_u_POW, axiom, ![V_x]: (~ (s(t_h4s_extreals_extreal,V_x) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)) => ?[V_n]: p(s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_pow(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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_extreals_supu_u_eq, axiom, ![V_x, V_p]: (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))))) & ![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_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_lebesgues_lemmau_u_fnu_u_1, axiom, ![TV_u_27a]: ![V_x, V_n, V_m, V_k, V_f]: ((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_bools_in(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_count(s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n)))))))) & (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_div(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_k))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_pow(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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n))))),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_lt(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_div(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_k))),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_pow(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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n))))))))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_fnu_u_seq(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_nums_num,V_n),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_div(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_k))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_pow(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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_lebesgues_lemmau_u_fnu_u_2, axiom, ![TV_u_27a]: ![V_x, V_n, V_m, V_f]: ((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_pow(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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n))),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_fnu_u_seq(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_nums_num,V_n),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_pow(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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n))))).
fof(ah4s_lebesgues_lemmau_u_fnu_u_3, axiom, ![TV_u_27a]: ![V_x, V_n, V_m, V_f]: ((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_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_pow(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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n))),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_k]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_count(s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n)))))))) & (p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_div(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_k))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_pow(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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n))))),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_lt(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_div(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_k))),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_pow(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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,V_n))))))))))))).
fof(ah4s_lebesgues_lemmau_u_fnu_u_5, axiom, ![TV_u_27a]: ![V_x, V_n, V_m, V_f]: (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_fnu_u_seq(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_nums_num,V_n),s(TV_u_27a,V_x)))))))).
fof(ah4s_lebesgues_lemmau_u_fnu_u_monou_u_increasing, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_m, V_f, 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(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,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(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_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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(TV_u_27a,V_x))),s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,h4s_lebesgues_fnu_u_seq(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_nums_num,V_n),s(TV_u_27a,V_x))) => ![V_x, V_m, V_f]: (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_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(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,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(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_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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(TV_u_27a,V_x))))))))).
fof(ah4s_lebesgues_lemmau_u_fnu_u_upperu_u_bounded, axiom, ![TV_u_27a]: ![V_x, V_n, V_m, V_f]: (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_lebesgues_fnu_u_seq(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_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),V_f),s(TV_u_27a,V_x)))))))).
fof(ch4s_lebesgues_lemmau_u_fnu_u_sup, conjecture, ![TV_u_27a]: ![V_uu_0]: (![V_m, V_f, 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(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,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(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_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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(TV_u_27a,V_x))),s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,h4s_lebesgues_fnu_u_seq(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_nums_num,V_n),s(TV_u_27a,V_x))) => ![V_x, V_m, V_f]: ((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))))))) => 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(TV_u_27a,t_h4s_extreals_extreal),t_fun(TV_u_27a,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(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_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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(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)))))).
