%   ORIGINAL: h4/measure/BOREL__MEASURABLE__SETS3
% 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/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/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/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% 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/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/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_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% 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/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/K__DEF: h4/combin/K = (\x 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/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__eq_c3: !n. h4/arithmetic/BIT2 n = h4/arithmetic/ZERO <=> F
% 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/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pred__set/GSPEC__T: h4/pred__set/GSPEC (\x. h4/pair/_2C x T) = h4/pred__set/UNIV
% Assm: h4/real/REAL__ADD__RID: !x. h4/realax/real__add x (h4/real/real__of__num h4/num/0) = x
% 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__INJ: !n m. h4/real/real__of__num m = h4/real/real__of__num n <=> m = n
% Assm: h4/real/abs0: !x. h4/real/abs x = h4/bool/COND (h4/real/real__lte (h4/real/real__of__num h4/num/0) x) x (h4/realax/real__neg x)
% 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__HALF__BETWEEN_c0: h4/realax/real__lt (h4/real/real__of__num h4/num/0) (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))))
% Assm: h4/real/REAL__HALF__BETWEEN_c1: h4/realax/real__lt (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)))) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm: h4/real/REAL__HALF__BETWEEN_c2: h4/real/real__lte (h4/real/real__of__num h4/num/0) (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))))
% Assm: h4/real/pow__rat_c5: !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/neg__rat_c0: !y x. h4/realax/real__neg (h4/real/_2F x y) = h4/bool/COND (y = h4/real/real__of__num h4/num/0) (h4/realax/real__neg (h4/marker/unint (h4/real/_2F x y))) (h4/real/_2F (h4/realax/real__neg x) y)
% Assm: h4/real/neg__rat_c1: !y x. h4/real/_2F x (h4/realax/real__neg y) = h4/bool/COND (y = h4/real/real__of__num h4/num/0) (h4/marker/unint (h4/real/_2F x y)) (h4/real/_2F (h4/realax/real__neg x) y)
% Assm: h4/real/eq__ints_c0: !n m. h4/real/real__of__num n = h4/real/real__of__num m <=> n = m
% Assm: h4/seq/SEQ__CONST: !k. h4/seq/_2D_2D_3E (\x. k) k
% Assm: h4/seq/SEQ__ADD: !y0 y x0 x. h4/seq/_2D_2D_3E x x0 /\ h4/seq/_2D_2D_3E y y0 ==> h4/seq/_2D_2D_3E (\n. h4/realax/real__add (x n) (y n)) (h4/realax/real__add x0 y0)
% Assm: h4/seq/SEQ__POWER: !c. h4/realax/real__lt (h4/real/abs c) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) ==> h4/seq/_2D_2D_3E (\n. h4/real/pow c n) (h4/real/real__of__num h4/num/0)
% Assm: h4/seq/LE__SEQ__IMP__LE__LIM: !y x f. (!n. h4/real/real__lte x (f n)) /\ h4/seq/_2D_2D_3E f y ==> h4/real/real__lte x y
% 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/IN__BIGINTER__IMAGE: !x s f. h4/bool/IN x (h4/pred__set/BIGINTER (h4/pred__set/IMAGE f s)) <=> (!y. h4/bool/IN y s ==> h4/bool/IN x (f y))
% Assm: h4/extreal/extreal__nchotomy: !ee. ee = h4/extreal/NegInf \/ ee = h4/extreal/PosInf \/ (?r. ee = h4/extreal/Normal r)
% 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__add__def_c1: !a. h4/extreal/extreal__add h4/extreal/PosInf a = h4/extreal/PosInf
% Assm: h4/extreal/extreal__add__def_c2: h4/extreal/extreal__add h4/extreal/NegInf h4/extreal/PosInf = h4/extreal/PosInf
% Assm: h4/extreal/extreal__add__def_c3: !v2. h4/extreal/extreal__add (h4/extreal/Normal v2) h4/extreal/PosInf = h4/extreal/PosInf
% Assm: h4/extreal/extreal__add__def_c4: h4/extreal/extreal__add h4/extreal/NegInf h4/extreal/NegInf = h4/extreal/NegInf
% Assm: h4/extreal/extreal__add__def_c5: !v5. h4/extreal/extreal__add h4/extreal/NegInf (h4/extreal/Normal v5) = h4/extreal/NegInf
% Assm: h4/extreal/extreal__add__def_c6: !v3. h4/extreal/extreal__add (h4/extreal/Normal v3) h4/extreal/NegInf = h4/extreal/NegInf
% 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__lt__def: !y x. h4/extreal/extreal__lt x y <=> ~h4/extreal/extreal__le y x
% Assm: h4/extreal/extreal__ainv__def_c0: h4/extreal/extreal__ainv h4/extreal/NegInf = h4/extreal/PosInf
% Assm: h4/extreal/extreal__ainv__def_c1: h4/extreal/extreal__ainv h4/extreal/PosInf = h4/extreal/NegInf
% Assm: h4/extreal/extreal__ainv__def_c2: !x. h4/extreal/extreal__ainv (h4/extreal/Normal x) = h4/extreal/Normal (h4/realax/real__neg x)
% 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_c1: !x. ~(h4/extreal/Normal x = h4/extreal/PosInf)
% 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_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_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/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__neg: !y x. h4/extreal/extreal__lt (h4/extreal/extreal__ainv x) (h4/extreal/extreal__ainv y) <=> h4/extreal/extreal__lt y x
% 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/add__rzero: !x. h4/extreal/extreal__add x (h4/extreal/extreal__of__num h4/num/0) = x
% Assm: h4/extreal/neg__neg: !x. h4/extreal/extreal__ainv (h4/extreal/extreal__ainv x) = x
% Assm: h4/extreal/SIMP__EXTREAL__ARCH: !x. ~(x = h4/extreal/PosInf) ==> (?n. h4/extreal/extreal__le x (h4/extreal/extreal__of__num n))
% Assm: h4/measure/sigma__algebra__def: !a. h4/measure/sigma__algebra a <=> h4/measure/algebra a /\ (!c. h4/util__prob/countable c /\ h4/pred__set/SUBSET c (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/BIGUNION c) (h4/measure/subsets a))
% Assm: h4/measure/ALGEBRA__SPACE: !a. h4/measure/algebra a ==> h4/bool/IN (h4/measure/space a) (h4/measure/subsets a)
% Assm: h4/measure/SIGMA__ALGEBRA__FN__BIGINTER: !a. h4/measure/sigma__algebra a ==> h4/measure/subset__class (h4/measure/space a) (h4/measure/subsets a) /\ h4/bool/IN h4/pred__set/EMPTY (h4/measure/subsets a) /\ (!s. h4/bool/IN s (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/DIFF (h4/measure/space a) s) (h4/measure/subsets a)) /\ (!f. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (h4/measure/subsets a)) ==> h4/bool/IN (h4/pred__set/BIGINTER (h4/pred__set/IMAGE f h4/pred__set/UNIV)) (h4/measure/subsets a))
% Assm: h4/measure/SIGMA__ALGEBRA__BOREL: h4/measure/sigma__algebra h4/measure/Borel
% Assm: h4/measure/SPACE__BOREL: h4/measure/space h4/measure/Borel = h4/pred__set/UNIV
% Assm: h4/measure/BOREL__MEASURABLE__SETS1: !c. h4/bool/IN (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__lt x c))) (h4/measure/subsets h4/measure/Borel)
% Goal: !c. h4/bool/IN (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/extreal/extreal__le x c))) (h4/measure/subsets h4/measure/Borel)
%   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_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_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_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% 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_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_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_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% 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_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_Ku_u_DEF]: !x x. h4/combin/K 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_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_equ_c3]: !n. h4/arithmetic/BIT2 n = h4/arithmetic/ZERO <=> F
% 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_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_GSPECu_u_T]: !_0. (!x. happ _0 x = h4/pair/_2C x T) ==> h4/pred__set/GSPEC _0 = h4/pred__set/UNIV
% Assm [h4s_reals_REALu_u_ADDu_u_RID]: !x. h4/realax/real__add x (h4/real/real__of__num h4/num/0) = x
% 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_INJ]: !n m. h4/real/real__of__num m = h4/real/real__of__num n <=> m = n
% Assm [h4s_reals_abs0]: !x. h4/real/abs x = h4/bool/COND (h4/real/real__lte (h4/real/real__of__num h4/num/0) x) x (h4/realax/real__neg x)
% 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_HALFu_u_BETWEENu_c0]: h4/realax/real__lt (h4/real/real__of__num h4/num/0) (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))))
% Assm [h4s_reals_REALu_u_HALFu_u_BETWEENu_c1]: h4/realax/real__lt (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)))) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_reals_REALu_u_HALFu_u_BETWEENu_c2]: h4/real/real__lte (h4/real/real__of__num h4/num/0) (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))))
% Assm [h4s_reals_powu_u_ratu_c5]: !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_negu_u_ratu_c0]: !y x. ?v. (v <=> y = h4/real/real__of__num h4/num/0) /\ h4/realax/real__neg (h4/real/_2F x y) = h4/bool/COND v (h4/realax/real__neg (h4/marker/unint (h4/real/_2F x y))) (h4/real/_2F (h4/realax/real__neg x) y)
% Assm [h4s_reals_negu_u_ratu_c1]: !y x. ?v. (v <=> y = h4/real/real__of__num h4/num/0) /\ h4/real/_2F x (h4/realax/real__neg y) = h4/bool/COND v (h4/marker/unint (h4/real/_2F x y)) (h4/real/_2F (h4/realax/real__neg x) y)
% 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_seqs_SEQu_u_CONST]: !_0. (!k x. happ (happ _0 k) x = k) ==> (!k. h4/seq/_2D_2D_3E (happ _0 k) k)
% Assm [h4s_seqs_SEQu_u_ADD]: !_0. (!x y n. happ (happ (happ _0 x) y) n = h4/realax/real__add (happ x n) (happ y n)) ==> (!y0 y x0 x. h4/seq/_2D_2D_3E x x0 /\ h4/seq/_2D_2D_3E y y0 ==> h4/seq/_2D_2D_3E (happ (happ _0 x) y) (h4/realax/real__add x0 y0))
% Assm [h4s_seqs_SEQu_u_POWER]: !_0. (!c n. happ (happ _0 c) n = h4/real/pow c n) ==> (!c. h4/realax/real__lt (h4/real/abs c) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) ==> h4/seq/_2D_2D_3E (happ _0 c) (h4/real/real__of__num h4/num/0))
% Assm [h4s_seqs_LEu_u_SEQu_u_IMPu_u_LEu_u_LIM]: !y x f. (!n. h4/real/real__lte x (happ f n)) /\ h4/seq/_2D_2D_3E f y ==> h4/real/real__lte x y
% 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_INu_u_BIGINTERu_u_IMAGE]: !x s f. h4/bool/IN x (h4/pred__set/BIGINTER (h4/pred__set/IMAGE f s)) <=> (!y. h4/bool/IN y s ==> h4/bool/IN x (happ f y))
% Assm [h4s_extreals_extrealu_u_nchotomy]: !ee. ee = h4/extreal/NegInf \/ ee = h4/extreal/PosInf \/ (?r. ee = h4/extreal/Normal r)
% 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_addu_u_defu_c1]: !a. h4/extreal/extreal__add h4/extreal/PosInf a = h4/extreal/PosInf
% Assm [h4s_extreals_extrealu_u_addu_u_defu_c2]: h4/extreal/extreal__add h4/extreal/NegInf h4/extreal/PosInf = h4/extreal/PosInf
% Assm [h4s_extreals_extrealu_u_addu_u_defu_c3]: !v2. h4/extreal/extreal__add (h4/extreal/Normal v2) h4/extreal/PosInf = h4/extreal/PosInf
% Assm [h4s_extreals_extrealu_u_addu_u_defu_c4]: h4/extreal/extreal__add h4/extreal/NegInf h4/extreal/NegInf = h4/extreal/NegInf
% Assm [h4s_extreals_extrealu_u_addu_u_defu_c5]: !v5. h4/extreal/extreal__add h4/extreal/NegInf (h4/extreal/Normal v5) = h4/extreal/NegInf
% Assm [h4s_extreals_extrealu_u_addu_u_defu_c6]: !v3. h4/extreal/extreal__add (h4/extreal/Normal v3) h4/extreal/NegInf = h4/extreal/NegInf
% 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_ltu_u_def]: !y x. h4/extreal/extreal__lt x y <=> ~h4/extreal/extreal__le y x
% Assm [h4s_extreals_extrealu_u_ainvu_u_defu_c0]: h4/extreal/extreal__ainv h4/extreal/NegInf = h4/extreal/PosInf
% Assm [h4s_extreals_extrealu_u_ainvu_u_defu_c1]: h4/extreal/extreal__ainv h4/extreal/PosInf = h4/extreal/NegInf
% Assm [h4s_extreals_extrealu_u_ainvu_u_defu_c2]: !x. h4/extreal/extreal__ainv (h4/extreal/Normal x) = h4/extreal/Normal (h4/realax/real__neg x)
% 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_c1]: !x. ~(h4/extreal/Normal x = h4/extreal/PosInf)
% 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_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_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_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_neg]: !y x. h4/extreal/extreal__lt (h4/extreal/extreal__ainv x) (h4/extreal/extreal__ainv y) <=> h4/extreal/extreal__lt y x
% 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_addu_u_rzero]: !x. h4/extreal/extreal__add x (h4/extreal/extreal__of__num h4/num/0) = x
% Assm [h4s_extreals_negu_u_neg]: !x. h4/extreal/extreal__ainv (h4/extreal/extreal__ainv x) = x
% Assm [h4s_extreals_SIMPu_u_EXTREALu_u_ARCH]: !x. ~(x = h4/extreal/PosInf) ==> (?n. h4/extreal/extreal__le x (h4/extreal/extreal__of__num n))
% Assm [h4s_measures_sigmau_u_algebrau_u_def]: !a. h4/measure/sigma__algebra a <=> h4/measure/algebra a /\ (!c. h4/util__prob/countable c /\ h4/pred__set/SUBSET c (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/BIGUNION c) (h4/measure/subsets a))
% Assm [h4s_measures_ALGEBRAu_u_SPACE]: !a. h4/measure/algebra a ==> h4/bool/IN (h4/measure/space a) (h4/measure/subsets a)
% Assm [h4s_measures_SIGMAu_u_ALGEBRAu_u_FNu_u_BIGINTER]: !a. h4/measure/sigma__algebra a ==> h4/measure/subset__class (h4/measure/space a) (h4/measure/subsets a) /\ h4/bool/IN h4/pred__set/EMPTY (h4/measure/subsets a) /\ (!s. h4/bool/IN s (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/DIFF (h4/measure/space a) s) (h4/measure/subsets a)) /\ (!f. h4/bool/IN f (h4/util__prob/FUNSET h4/pred__set/UNIV (h4/measure/subsets a)) ==> h4/bool/IN (h4/pred__set/BIGINTER (h4/pred__set/IMAGE f h4/pred__set/UNIV)) (h4/measure/subsets a))
% Assm [h4s_measures_SIGMAu_u_ALGEBRAu_u_BOREL]: h4/measure/sigma__algebra h4/measure/Borel
% Assm [h4s_measures_SPACEu_u_BOREL]: h4/measure/space h4/measure/Borel = h4/pred__set/UNIV
% Assm [h4s_measures_BORELu_u_MEASURABLEu_u_SETS1]: !_0. (!c x. happ (happ _0 c) x = h4/pair/_2C x (h4/extreal/extreal__lt x c)) ==> (!c. h4/bool/IN (h4/pred__set/GSPEC (happ _0 c)) (h4/measure/subsets h4/measure/Borel))
% Goal: !_0. (!c x. happ (happ _0 c) x = h4/pair/_2C x (h4/extreal/extreal__le x c)) ==> (!c. h4/bool/IN (h4/pred__set/GSPEC (happ _0 c)) (h4/measure/subsets h4/measure/Borel))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q114360,TV_Q114356]: ![V_f, V_g]: (![V_x]: s(TV_Q114356,happ(s(t_fun(TV_Q114360,TV_Q114356),V_f),s(TV_Q114360,V_x))) = s(TV_Q114356,happ(s(t_fun(TV_Q114360,TV_Q114356),V_g),s(TV_Q114360,V_x))) => s(t_fun(TV_Q114360,TV_Q114356),V_f) = s(t_fun(TV_Q114360,TV_Q114356),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,f)) => 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_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_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_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_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_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_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_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_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_combins_Ku_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27a,h4s_combins_k(s(TV_u_27a,V_x),s(TV_u_27b,V_x0))) = s(TV_u_27a,V_x)).
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_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_equ_c1, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero) <=> p(s(t_bool,f)))).
fof(ah4s_numerals_numeralu_u_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,f)))).
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_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_GSPECu_u_T, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,t))) => s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_uu_0))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))).
fof(ah4s_reals_REALu_u_ADDu_u_RID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(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_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_INJ, axiom, ![V_n, V_m]: (s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_reals_abs0, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_bools_cond(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))),s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x)))))).
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_HALFu_u_BETWEENu_c0, axiom, 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,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))))))))))))).
fof(ah4s_reals_REALu_u_HALFu_u_BETWEENu_c1, axiom, p(s(t_bool,h4s_realaxs_realu_u_lt(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_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_HALFu_u_BETWEENu_c2, axiom, 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_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))))))))))))).
fof(ah4s_reals_powu_u_ratu_c5, 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_negu_u_ratu_c0, axiom, ![V_y, V_x]: ?[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_realaxs_real,h4s_realaxs_realu_u_neg(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_bools_cond(s(t_bool,V_v),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,h4s_markers_unint(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_reals_u_2f(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_y))))))).
fof(ah4s_reals_negu_u_ratu_c1, axiom, ![V_y, V_x]: ?[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_realaxs_real,h4s_reals_u_2f(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_y))))) = s(t_h4s_realaxs_real,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_realaxs_real,h4s_markers_unint(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_reals_u_2f(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_y))))))).
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_seqs_SEQu_u_CONST, axiom, ![V_uu_0]: (![V_k, V_x]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_realaxs_real,V_k))),s(t_h4s_nums_num,V_x))) = s(t_h4s_realaxs_real,V_k) => ![V_k]: p(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_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_realaxs_real,V_k))),s(t_h4s_realaxs_real,V_k)))))).
fof(ah4s_seqs_SEQu_u_ADD, 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_realaxs_realu_u_add(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_u_2du_2du_3e(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_u_2du_2du_3e(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_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)),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_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x0),s(t_h4s_realaxs_real,V_y0))))))))).
fof(ah4s_seqs_SEQu_u_POWER, axiom, ![V_uu_0]: (![V_c, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_realaxs_real,V_c))),s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_c),s(t_h4s_nums_num,V_n))) => ![V_c]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_c))),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)))))))))) => p(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_h4s_realaxs_real,t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_realaxs_real,V_c))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))))))).
fof(ah4s_seqs_LEu_u_SEQu_u_IMPu_u_LEu_u_LIM, axiom, ![V_y, V_x, V_f]: ((![V_n]: p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),s(t_h4s_nums_num,V_n)))))) & p(s(t_bool,h4s_seqs_u_2du_2du_3e(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_f),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_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_INu_u_BIGINTERu_u_IMAGE, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_biginter(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),V_f),s(t_fun(TV_u_27b,t_bool),V_s)))))))) <=> ![V_y]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,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),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_y))))))))).
fof(ah4s_extreals_extrealu_u_nchotomy, axiom, ![V_ee]: (s(t_h4s_extreals_extreal,V_ee) = s(t_h4s_extreals_extreal,h4s_extreals_neginf) | (s(t_h4s_extreals_extreal,V_ee) = s(t_h4s_extreals_extreal,h4s_extreals_posinf) | ?[V_r]: s(t_h4s_extreals_extreal,V_ee) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_r)))))).
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_addu_u_defu_c1, axiom, ![V_a]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,V_a))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)).
fof(ah4s_extreals_extrealu_u_addu_u_defu_c2, axiom, s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,h4s_extreals_posinf))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)).
fof(ah4s_extreals_extrealu_u_addu_u_defu_c3, axiom, ![V_v2]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(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_h4s_extreals_extreal,h4s_extreals_posinf)).
fof(ah4s_extreals_extrealu_u_addu_u_defu_c4, axiom, s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,h4s_extreals_neginf))) = s(t_h4s_extreals_extreal,h4s_extreals_neginf)).
fof(ah4s_extreals_extrealu_u_addu_u_defu_c5, axiom, ![V_v5]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_v5))))) = s(t_h4s_extreals_extreal,h4s_extreals_neginf)).
fof(ah4s_extreals_extrealu_u_addu_u_defu_c6, axiom, ![V_v3]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_add(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_h4s_extreals_extreal,h4s_extreals_neginf)).
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_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_ainvu_u_defu_c0, axiom, s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ainv(s(t_h4s_extreals_extreal,h4s_extreals_neginf))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)).
fof(ah4s_extreals_extrealu_u_ainvu_u_defu_c1, axiom, s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ainv(s(t_h4s_extreals_extreal,h4s_extreals_posinf))) = s(t_h4s_extreals_extreal,h4s_extreals_neginf)).
fof(ah4s_extreals_extrealu_u_ainvu_u_defu_c2, axiom, ![V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ainv(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,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x)))))).
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_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_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_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_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_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_neg, axiom, ![V_y, V_x]: s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ainv(s(t_h4s_extreals_extreal,V_x))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ainv(s(t_h4s_extreals_extreal,V_y))))) = s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_x)))).
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_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_negu_u_neg, axiom, ![V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ainv(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ainv(s(t_h4s_extreals_extreal,V_x))))) = s(t_h4s_extreals_extreal,V_x)).
fof(ah4s_extreals_SIMPu_u_EXTREALu_u_ARCH, 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_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_measures_sigmau_u_algebrau_u_def, axiom, ![TV_u_27a]: ![V_a]: (p(s(t_bool,h4s_measures_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))) <=> (p(s(t_bool,h4s_measures_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))) & ![V_c]: ((p(s(t_bool,h4s_utilu_u_probs_countable(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_c)))) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_c),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_c))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))))))).
fof(ah4s_measures_ALGEBRAu_u_SPACE, axiom, ![TV_u_27a]: ![V_a]: (p(s(t_bool,h4s_measures_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))))).
fof(ah4s_measures_SIGMAu_u_ALGEBRAu_u_FNu_u_BIGINTER, axiom, ![TV_u_27a]: ![V_a]: (p(s(t_bool,h4s_measures_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))) => (p(s(t_bool,h4s_measures_subsetu_u_class(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))) & (![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))))))) & ![V_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_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_biginter(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_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))))))))).
fof(ah4s_measures_SIGMAu_u_ALGEBRAu_u_BOREL, axiom, p(s(t_bool,h4s_measures_sigmau_u_algebra(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_SPACEu_u_BOREL, axiom, s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_measures_space(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_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_univ)).
fof(ah4s_measures_BORELu_u_MEASURABLEu_u_SETS1, axiom, ![V_uu_0]: (![V_c, V_x]: 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_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool))),V_uu_0),s(t_h4s_extreals_extreal,V_c))),s(t_h4s_extreals_extreal,V_x))) = s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool),h4s_pairs_u_2c(s(t_h4s_extreals_extreal,V_x),s(t_bool,h4s_extreals_extrealu_u_lt(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_c))))) => ![V_c]: p(s(t_bool,h4s_bools_in(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_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool))),V_uu_0),s(t_h4s_extreals_extreal,V_c))))),s(t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool),h4s_measures_subsets(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(ch4s_measures_BORELu_u_MEASURABLEu_u_SETS3, conjecture, ![V_uu_0]: (![V_c, V_x]: 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_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool))),V_uu_0),s(t_h4s_extreals_extreal,V_c))),s(t_h4s_extreals_extreal,V_x))) = s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool),h4s_pairs_u_2c(s(t_h4s_extreals_extreal,V_x),s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_c))))) => ![V_c]: p(s(t_bool,h4s_bools_in(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_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_h4s_pairs_prod(t_h4s_extreals_extreal,t_bool))),V_uu_0),s(t_h4s_extreals_extreal,V_c))))),s(t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool),h4s_measures_subsets(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)))))))).
