%   ORIGINAL: h4/lebesgue/finite__space__integral__reduce
% 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/lebesgue/finite__space__integral__def: !m f. h4/lebesgue/finite__space__integral m f = h4/extreal/EXTREAL__SUM__IMAGE (\r. h4/extreal/extreal__mul r (h4/extreal/Normal (h4/measure/measure m (h4/pred__set/INTER (h4/util__prob/PREIMAGE f (h4/pred__set/INSERT r h4/pred__set/EMPTY)) (h4/measure/m__space m))))) (h4/pred__set/IMAGE f (h4/measure/m__space m))
% Assm: h4/measure/finite__additivity__sufficient__for__finite__spaces2: !m. h4/measure/sigma__algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/pred__set/FINITE (h4/measure/m__space m) /\ h4/measure/positive m /\ h4/measure/additive m ==> h4/measure/measure__space m
% Assm: h4/lebesgue/measurable__sequence: !m f. h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> (?fi ri. (!x. h4/extreal/ext__mono__increasing (\i. fi i x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. fi i x) h4/pred__set/UNIV) = h4/measure/fn__plus f x) /\ (!i. h4/bool/IN (ri i) (h4/lebesgue/psfis m (fi i))) /\ (!i x. h4/extreal/extreal__le (fi i x) (h4/measure/fn__plus f x)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (fi i x)) /\ h4/lebesgue/pos__fn__integral m (h4/measure/fn__plus f) = h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. h4/lebesgue/pos__fn__integral m (fi i)) h4/pred__set/UNIV)) /\ (?gi vi. (!x. h4/extreal/ext__mono__increasing (\i. gi i x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. gi i x) h4/pred__set/UNIV) = h4/measure/fn__minus f x) /\ (!i. h4/bool/IN (vi i) (h4/lebesgue/psfis m (gi i))) /\ (!i x. h4/extreal/extreal__le (gi i x) (h4/measure/fn__minus f x)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (gi i x)) /\ h4/lebesgue/pos__fn__integral m (h4/measure/fn__minus f) = h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. h4/lebesgue/pos__fn__integral m (gi i)) h4/pred__set/UNIV))
% Assm: h4/lebesgue/integrable__not__infty: !m f. h4/measure/measure__space m /\ h4/lebesgue/integrable m f /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) ==> (?g. h4/lebesgue/integrable m g /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (g x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> ~(g x = h4/extreal/PosInf)) /\ h4/lebesgue/integral m f = h4/lebesgue/integral m g)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/measure/space__def: !y x. h4/measure/space (h4/pair/_2C x y) = x
% Assm: h4/measure/subsets__def: !y x. h4/measure/subsets (h4/pair/_2C x y) = y
% Assm: h4/measure/finite__additivity__sufficient__for__finite__spaces: !s m. h4/measure/sigma__algebra s /\ h4/pred__set/FINITE (h4/measure/space s) /\ h4/measure/positive (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) m)) /\ h4/measure/additive (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) m)) ==> h4/measure/measure__space (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) m))
% Assm: h4/measure/MEASURE__SPACE__REDUCE: !m. h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/measure/measurable__sets m) (h4/measure/measure m)) = m
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/measure/FN__PLUS__POS: !x g. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/measure/fn__plus g x)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% 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)
% 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/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/lebesgue/integral__sequence: !m f. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) /\ h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> h4/lebesgue/pos__fn__integral m f = h4/extreal/extreal__sup (h4/pred__set/IMAGE (\i. h4/lebesgue/pos__fn__integral m (h4/lebesgue/fn__seq m f i)) h4/pred__set/UNIV)
% Assm: h4/lebesgue/lemma__fn__in__psfis: !n m f. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f x)) /\ h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> h4/bool/IN (h4/lebesgue/fn__seq__integral m f n) (h4/lebesgue/psfis m (h4/lebesgue/fn__seq m f n))
% Assm: h4/measure/IN__MEASURABLE__BOREL__FN__MINUS: !f a. h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) ==> h4/bool/IN (h4/measure/fn__minus f) (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/measure/IN__MEASURABLE__BOREL__FN__PLUS: !f a. h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) ==> h4/bool/IN (h4/measure/fn__plus f) (h4/measure/measurable a h4/measure/Borel)
% Assm: h4/lebesgue/lemma__fn__sup: !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
% Assm: h4/measure/FN__MINUS__POS: !x g. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (h4/measure/fn__minus g x)
% Assm: h4/measure/indicator__fn__def: !s. h4/measure/indicator__fn s = (\x. h4/bool/COND (h4/bool/IN x s) (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/extreal/extreal__of__num h4/num/0))
% Assm: h4/measure/measure__space__def: !m. h4/measure/measure__space m <=> h4/measure/sigma__algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/countably__additive m
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/extreal/add__rzero: !x. h4/extreal/extreal__add x (h4/extreal/extreal__of__num h4/num/0) = x
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/extreal/lt__infty_c6: !x. ~(x = h4/extreal/NegInf) <=> h4/extreal/extreal__lt h4/extreal/NegInf x
% Assm: h4/extreal/lt__infty_c5: !x. ~(x = h4/extreal/PosInf) <=> h4/extreal/extreal__lt x h4/extreal/PosInf
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/extreal/lt__infty_c1: !y. h4/extreal/extreal__lt (h4/extreal/Normal y) h4/extreal/PosInf
% Assm: h4/measure/null__set__def: !s m. h4/measure/null__set m s <=> h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/measure/measure m s = h4/real/real__of__num h4/num/0
% 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/lt__infty_c4: !x. ~h4/extreal/extreal__lt h4/extreal/PosInf x
% Assm: h4/extreal/lt__infty_c0: !y. h4/extreal/extreal__lt h4/extreal/NegInf (h4/extreal/Normal y)
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/extreal/sub__refl: !x. ~(x = h4/extreal/NegInf) /\ ~(x = h4/extreal/PosInf) ==> h4/extreal/extreal__sub x x = h4/extreal/extreal__of__num h4/num/0
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/extreal/mul__rone: !x. h4/extreal/extreal__mul x (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/extreal/le__trans: !z y x. h4/extreal/extreal__le x y /\ h4/extreal/extreal__le y z ==> h4/extreal/extreal__le x z
% Assm: h4/bool/CONJ__SYM: !t2 t1. t1 /\ t2 <=> t2 /\ t1
% 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/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/le__infty_c3: !x. h4/extreal/extreal__le h4/extreal/PosInf x <=> x = h4/extreal/PosInf
% Assm: h4/measure/MEASURE__SPACE__UNION: !t s m. h4/measure/measure__space m /\ h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/bool/IN t (h4/measure/measurable__sets m) ==> h4/bool/IN (h4/pred__set/UNION s t) (h4/measure/measurable__sets m)
% Assm: h4/extreal/extreal__11: !a_27 a. h4/extreal/Normal a = h4/extreal/Normal a_27 <=> a = a_27
% Assm: h4/extreal/num__not__infty_c0: !n. ~(h4/extreal/extreal__of__num n = h4/extreal/NegInf)
% Assm: h4/extreal/extreal__not__infty_c0: !x. ~(h4/extreal/Normal x = h4/extreal/NegInf)
% Assm: h4/extreal/le__refl: !x. h4/extreal/extreal__le x x
% Assm: h4/extreal/mul__rzero: !x. h4/extreal/extreal__mul x (h4/extreal/extreal__of__num h4/num/0) = h4/extreal/extreal__of__num h4/num/0
% Assm: h4/extreal/le__infty_c0: !x. h4/extreal/extreal__le h4/extreal/NegInf x
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/extreal/num__not__infty_c1: !n. ~(h4/extreal/extreal__of__num n = h4/extreal/PosInf)
% Assm: h4/extreal/extreal__not__infty_c1: !x. ~(h4/extreal/Normal x = h4/extreal/PosInf)
% Assm: h4/extreal/le__infty_c1: !x. h4/extreal/extreal__le x h4/extreal/PosInf
% 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__sub__def_c1: !a. h4/extreal/extreal__sub h4/extreal/PosInf a = h4/extreal/PosInf
% 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__add__def_c4: h4/extreal/extreal__add h4/extreal/NegInf h4/extreal/NegInf = h4/extreal/NegInf
% Assm: h4/extreal/extreal__sub__def_c6: !v3. h4/extreal/extreal__sub (h4/extreal/Normal v3) h4/extreal/NegInf = h4/extreal/PosInf
% 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_c3: !v2. h4/extreal/extreal__add (h4/extreal/Normal v2) h4/extreal/PosInf = h4/extreal/PosInf
% 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/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/measure/MEASURE__SPACE__EMPTY__MEASURABLE: !m. h4/measure/measure__space m ==> h4/bool/IN h4/pred__set/EMPTY (h4/measure/measurable__sets m)
% Assm: h4/extreal/le__infty_c2: !x. h4/extreal/extreal__le x h4/extreal/NegInf <=> x = h4/extreal/NegInf
% Assm: h4/extreal/extreal__lt__def: !y x. h4/extreal/extreal__lt x y <=> ~h4/extreal/extreal__le y x
% Assm: h4/extreal/extreal__add__def_c2: h4/extreal/extreal__add h4/extreal/NegInf h4/extreal/PosInf = h4/extreal/PosInf
% 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/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/real/REAL__EQ__SUB__LADD: !z y x. x = h4/real/real__sub y z <=> h4/realax/real__add x z = 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/real/REAL__ADD__COMM: !y x. h4/realax/real__add x y = h4/realax/real__add y x
% Assm: h4/pred__set/UNION__OVER__INTER: !u t s. h4/pred__set/INTER s (h4/pred__set/UNION t u) = h4/pred__set/UNION (h4/pred__set/INTER s t) (h4/pred__set/INTER s u)
% Assm: h4/extreal/extreal__distinct_c0: ~(h4/extreal/NegInf = h4/extreal/PosInf)
% Assm: h4/pred__set/IN__DIFF: !x t s. h4/bool/IN x (h4/pred__set/DIFF s t) <=> h4/bool/IN x s /\ ~h4/bool/IN x t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/pred__set/INTER__COMM: !t s. h4/pred__set/INTER s t = h4/pred__set/INTER t s
% Assm: h4/pred__set/INTER__EMPTY_c0: !s. h4/pred__set/INTER h4/pred__set/EMPTY s = h4/pred__set/EMPTY
% Assm: h4/pred__set/IN__INTER: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/extreal/extreal__distinct_c1: !a. ~(h4/extreal/NegInf = h4/extreal/Normal a)
% Assm: h4/pred__set/INTER__EMPTY_c1: !s. h4/pred__set/INTER s h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm: h4/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% 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/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/combin/S__DEF: h4/combin/S = (\f g x. f x (g x))
% Assm: h4/combin/C__DEF: h4/combin/C = (\f x y. f y x)
% Assm: h4/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% Assm: h4/extreal/extreal__distinct_c2: !a. ~(h4/extreal/PosInf = h4/extreal/Normal a)
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/dc__cond: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Goal: !m f. h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> ~(f x = h4/extreal/NegInf) /\ ~(f x = h4/extreal/PosInf)) /\ h4/pred__set/FINITE (h4/measure/m__space m) ==> h4/lebesgue/integral m f = h4/lebesgue/finite__space__integral m f
%   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_lebesgues_finiteu_u_spaceu_u_integralu_u_def]: !_0. (!f m r. happ (happ (happ _0 f) m) r = h4/extreal/extreal__mul r (h4/extreal/Normal (happ (h4/measure/measure m) (h4/pred__set/INTER (h4/util__prob/PREIMAGE f (h4/pred__set/INSERT r h4/pred__set/EMPTY)) (h4/measure/m__space m))))) ==> (!m f. h4/lebesgue/finite__space__integral m f = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ _0 f) m) (h4/pred__set/IMAGE f (h4/measure/m__space m)))
% Assm [h4s_measures_finiteu_u_additivityu_u_sufficientu_u_foru_u_finiteu_u_spaces2]: !m. h4/measure/sigma__algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/pred__set/FINITE (h4/measure/m__space m) /\ h4/measure/positive m /\ h4/measure/additive m ==> h4/measure/measure__space m
% Assm [h4s_lebesgues_measurableu_u_sequence]: !_1. (!m fi i. happ (happ (happ _1 m) fi) i = h4/lebesgue/pos__fn__integral m (happ fi i)) ==> (!_0. (!fi x i. happ (happ (happ _0 fi) x) i = happ (happ fi i) x) ==> (!m f. h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> (?fi ri. (!x. h4/extreal/ext__mono__increasing (happ (happ _0 fi) x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 fi) x) h4/pred__set/UNIV) = happ (h4/measure/fn__plus f) x) /\ (!i. h4/bool/IN (happ ri i) (h4/lebesgue/psfis m (happ fi i))) /\ (!i x. h4/extreal/extreal__le (happ (happ fi i) x) (happ (h4/measure/fn__plus f) x)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (happ fi i) x)) /\ h4/lebesgue/pos__fn__integral m (h4/measure/fn__plus f) = h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _1 m) fi) h4/pred__set/UNIV)) /\ (?gi vi. (!x. h4/extreal/ext__mono__increasing (happ (happ _0 gi) x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 gi) x) h4/pred__set/UNIV) = happ (h4/measure/fn__minus f) x) /\ (!i. h4/bool/IN (happ vi i) (h4/lebesgue/psfis m (happ gi i))) /\ (!i x. h4/extreal/extreal__le (happ (happ gi i) x) (happ (h4/measure/fn__minus f) x)) /\ (!i x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (happ gi i) x)) /\ h4/lebesgue/pos__fn__integral m (h4/measure/fn__minus f) = h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _1 m) gi) h4/pred__set/UNIV))))
% Assm [h4s_lebesgues_integrableu_u_notu_u_infty]: !m f. h4/measure/measure__space m /\ h4/lebesgue/integrable m f /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) ==> (?g. h4/lebesgue/integrable m g /\ (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ g x)) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> ~(happ g x = h4/extreal/PosInf)) /\ h4/lebesgue/integral m f = h4/lebesgue/integral m g)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% 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_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_measures_spaceu_u_def]: !y x. h4/measure/space (h4/pair/_2C x y) = x
% Assm [h4s_measures_subsetsu_u_def]: !y x. h4/measure/subsets (h4/pair/_2C x y) = y
% Assm [h4s_measures_finiteu_u_additivityu_u_sufficientu_u_foru_u_finiteu_u_spaces]: !s m. h4/measure/sigma__algebra s /\ h4/pred__set/FINITE (h4/measure/space s) /\ h4/measure/positive (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) m)) /\ h4/measure/additive (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) m)) ==> h4/measure/measure__space (h4/pair/_2C (h4/measure/space s) (h4/pair/_2C (h4/measure/subsets s) m))
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_REDUCE]: !m. h4/pair/_2C (h4/measure/m__space m) (h4/pair/_2C (h4/measure/measurable__sets m) (h4/measure/measure m)) = m
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_measures_FNu_u_PLUSu_u_POS]: !x g. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (h4/measure/fn__plus g) x)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% 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 (happ (h4/lebesgue/fn__seq m f n) x) (happ f x)
% 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) (happ (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 = happ (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_combins_Iu_u_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_lebesgues_integralu_u_sequence]: !_0. (!m f i. happ (happ (happ _0 m) f) i = h4/lebesgue/pos__fn__integral m (h4/lebesgue/fn__seq m f i)) ==> (!m f. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) /\ h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> h4/lebesgue/pos__fn__integral m f = h4/extreal/extreal__sup (h4/pred__set/IMAGE (happ (happ _0 m) f) h4/pred__set/UNIV))
% Assm [h4s_lebesgues_lemmau_u_fnu_u_inu_u_psfis]: !n m f. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f x)) /\ h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) ==> h4/bool/IN (h4/lebesgue/fn__seq__integral m f n) (h4/lebesgue/psfis m (h4/lebesgue/fn__seq m f n))
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_FNu_u_MINUS]: !f a. h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) ==> h4/bool/IN (h4/measure/fn__minus f) (h4/measure/measurable a h4/measure/Borel)
% Assm [h4s_measures_INu_u_MEASURABLEu_u_BORELu_u_FNu_u_PLUS]: !f a. h4/bool/IN f (h4/measure/measurable a h4/measure/Borel) ==> h4/bool/IN (h4/measure/fn__plus f) (h4/measure/measurable a h4/measure/Borel)
% Assm [h4s_lebesgues_lemmau_u_fnu_u_sup]: !_0. (!m f x n. happ (happ (happ (happ _0 m) f) x) n = happ (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)
% Assm [h4s_measures_FNu_u_MINUSu_u_POS]: !x g. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ (h4/measure/fn__minus g) x)
% Assm [h4s_measures_indicatoru_u_fnu_u_def]: !s x. h4/measure/indicator__fn s x = h4/bool/COND (h4/bool/IN x s) (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/extreal/extreal__of__num h4/num/0)
% Assm [h4s_measures_measureu_u_spaceu_u_def]: !m. h4/measure/measure__space m <=> h4/measure/sigma__algebra (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) /\ h4/measure/positive m /\ h4/measure/countably__additive m
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_extreals_addu_u_rzero]: !x. h4/extreal/extreal__add x (h4/extreal/extreal__of__num h4/num/0) = x
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% 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_inftyu_c5]: !x. ~(x = h4/extreal/PosInf) <=> h4/extreal/extreal__lt x h4/extreal/PosInf
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_extreals_ltu_u_inftyu_c1]: !y. h4/extreal/extreal__lt (h4/extreal/Normal y) h4/extreal/PosInf
% Assm [h4s_measures_nullu_u_setu_u_def]: !s m. h4/measure/null__set m s <=> h4/bool/IN s (h4/measure/measurable__sets m) /\ happ (h4/measure/measure m) s = h4/real/real__of__num h4/num/0
% 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_ltu_u_inftyu_c4]: !x. ~h4/extreal/extreal__lt h4/extreal/PosInf x
% Assm [h4s_extreals_ltu_u_inftyu_c0]: !y. h4/extreal/extreal__lt h4/extreal/NegInf (h4/extreal/Normal y)
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_extreals_subu_u_refl]: !x. ~(x = h4/extreal/NegInf) /\ ~(x = h4/extreal/PosInf) ==> h4/extreal/extreal__sub x x = h4/extreal/extreal__of__num h4/num/0
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_extreals_mulu_u_rone]: !x. h4/extreal/extreal__mul x (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_extreals_leu_u_trans]: !z y x. h4/extreal/extreal__le x y /\ h4/extreal/extreal__le y z ==> h4/extreal/extreal__le x z
% Assm [h4s_bools_CONJu_u_SYM]: !t2 t1. t1 /\ t2 <=> t2 /\ t1
% 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_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_leu_u_inftyu_c3]: !x. h4/extreal/extreal__le h4/extreal/PosInf x <=> x = h4/extreal/PosInf
% Assm [h4s_measures_MEASUREu_u_SPACEu_u_UNION]: !t s m. h4/measure/measure__space m /\ h4/bool/IN s (h4/measure/measurable__sets m) /\ h4/bool/IN t (h4/measure/measurable__sets m) ==> h4/bool/IN (h4/pred__set/UNION s t) (h4/measure/measurable__sets m)
% Assm [h4s_extreals_extrealu_u_11]: !a_27 a. h4/extreal/Normal a = h4/extreal/Normal a_27 <=> a = a_27
% Assm [h4s_extreals_numu_u_notu_u_inftyu_c0]: !n. ~(h4/extreal/extreal__of__num n = h4/extreal/NegInf)
% Assm [h4s_extreals_extrealu_u_notu_u_inftyu_c0]: !x. ~(h4/extreal/Normal x = h4/extreal/NegInf)
% Assm [h4s_extreals_leu_u_refl]: !x. h4/extreal/extreal__le x x
% Assm [h4s_extreals_mulu_u_rzero]: !x. h4/extreal/extreal__mul x (h4/extreal/extreal__of__num h4/num/0) = h4/extreal/extreal__of__num h4/num/0
% Assm [h4s_extreals_leu_u_inftyu_c0]: !x. h4/extreal/extreal__le h4/extreal/NegInf x
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_extreals_numu_u_notu_u_inftyu_c1]: !n. ~(h4/extreal/extreal__of__num n = h4/extreal/PosInf)
% Assm [h4s_extreals_extrealu_u_notu_u_inftyu_c1]: !x. ~(h4/extreal/Normal x = h4/extreal/PosInf)
% Assm [h4s_extreals_leu_u_inftyu_c1]: !x. h4/extreal/extreal__le x h4/extreal/PosInf
% 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_subu_u_defu_c1]: !a. h4/extreal/extreal__sub h4/extreal/PosInf a = h4/extreal/PosInf
% 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_addu_u_defu_c4]: h4/extreal/extreal__add h4/extreal/NegInf h4/extreal/NegInf = h4/extreal/NegInf
% Assm [h4s_extreals_extrealu_u_subu_u_defu_c6]: !v3. h4/extreal/extreal__sub (h4/extreal/Normal v3) h4/extreal/NegInf = h4/extreal/PosInf
% 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_c3]: !v2. h4/extreal/extreal__add (h4/extreal/Normal v2) h4/extreal/PosInf = h4/extreal/PosInf
% 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_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_measures_MEASUREu_u_SPACEu_u_EMPTYu_u_MEASURABLE]: !m. h4/measure/measure__space m ==> h4/bool/IN h4/pred__set/EMPTY (h4/measure/measurable__sets m)
% Assm [h4s_extreals_leu_u_inftyu_c2]: !x. h4/extreal/extreal__le x h4/extreal/NegInf <=> x = h4/extreal/NegInf
% 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_addu_u_defu_c2]: h4/extreal/extreal__add h4/extreal/NegInf h4/extreal/PosInf = h4/extreal/PosInf
% 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_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_reals_REALu_u_EQu_u_SUBu_u_LADD]: !z y x. x = h4/real/real__sub y z <=> h4/realax/real__add x z = 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_reals_REALu_u_ADDu_u_COMM]: !y x. h4/realax/real__add x y = h4/realax/real__add y x
% Assm [h4s_predu_u_sets_UNIONu_u_OVERu_u_INTER]: !u t s. h4/pred__set/INTER s (h4/pred__set/UNION t u) = h4/pred__set/UNION (h4/pred__set/INTER s t) (h4/pred__set/INTER s u)
% Assm [h4s_extreals_extrealu_u_distinctu_c0]: ~(h4/extreal/NegInf = h4/extreal/PosInf)
% Assm [h4s_predu_u_sets_INu_u_DIFF]: !x t s. h4/bool/IN x (h4/pred__set/DIFF s t) <=> h4/bool/IN x s /\ ~h4/bool/IN x t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_predu_u_sets_INTERu_u_COMM]: !t s. h4/pred__set/INTER s t = h4/pred__set/INTER t s
% Assm [h4s_predu_u_sets_INTERu_u_EMPTYu_c0]: !s. h4/pred__set/INTER h4/pred__set/EMPTY s = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_INu_u_INTER]: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_extreals_extrealu_u_distinctu_c1]: !a. ~(h4/extreal/NegInf = h4/extreal/Normal a)
% Assm [h4s_predu_u_sets_INTERu_u_EMPTYu_c1]: !s. h4/pred__set/INTER s h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% 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_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_combins_Su_u_DEF]: !x x x. h4/combin/S x x x = happ (happ x x) (happ x x)
% Assm [h4s_combins_Cu_u_DEF]: !x x x. h4/combin/C x x x = happ (happ x x) x
% Assm [h4s_markers_Abbrevu_u_def]: !x. h4/marker/Abbrev x <=> x
% Assm [h4s_extreals_extrealu_u_distinctu_c2]: !a. ~(h4/extreal/PosInf = h4/extreal/Normal a)
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_dcu_u_cond]: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Goal: !m f. h4/measure/measure__space m /\ h4/bool/IN f (h4/measure/measurable (h4/pair/_2C (h4/measure/m__space m) (h4/measure/measurable__sets m)) h4/measure/Borel) /\ (!x. h4/bool/IN x (h4/measure/m__space m) ==> ~(happ f x = h4/extreal/NegInf) /\ ~(happ f x = h4/extreal/PosInf)) /\ h4/pred__set/FINITE (h4/measure/m__space m) ==> h4/lebesgue/integral m f = h4/lebesgue/finite__space__integral m f
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_Q1362801,TV_Q1362797]: ![V_f, V_g]: (![V_x]: s(TV_Q1362797,happ(s(t_fun(TV_Q1362801,TV_Q1362797),V_f),s(TV_Q1362801,V_x))) = s(TV_Q1362797,happ(s(t_fun(TV_Q1362801,TV_Q1362797),V_g),s(TV_Q1362801,V_x))) => s(t_fun(TV_Q1362801,TV_Q1362797),V_f) = s(t_fun(TV_Q1362801,TV_Q1362797),V_g))).
fof(ah4s_lebesgues_finiteu_u_spaceu_u_integralu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_m, V_r]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_extreals_extreal,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_h4s_extreals_extreal,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_h4s_extreals_extreal,V_r))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,V_r),s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_insert(s(t_h4s_extreals_extreal,V_r),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_empty))))),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))))))))))) => ![V_m, V_f]: s(t_h4s_extreals_extreal,h4s_lebesgues_finiteu_u_spaceu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(t_h4s_extreals_extreal,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_h4s_extreals_extreal,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))))).
fof(ah4s_measures_finiteu_u_additivityu_u_sufficientu_u_foru_u_finiteu_u_spaces2, axiom, ![TV_u_27a]: ![V_m]: ((p(s(t_bool,h4s_measures_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (p(s(t_bool,h4s_predu_u_sets_finite(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_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_measures_additive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))) => p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))).
fof(ah4s_lebesgues_measurableu_u_sequence, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_m, V_fi, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))))) => ![V_uu_0]: (![V_fi, V_x, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))) => ![V_m, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel))))))) => (?[V_fi, V_ri]: (![V_x]: p(s(t_bool,h4s_extreals_extu_u_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x)))))) & (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))) = s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x)))) & (![V_i]: p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_ri),s(t_h4s_nums_num,V_i))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i)))))))) & (![V_i, V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x)))))) & (![V_i, V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x)))))) & s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_fi))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ)))))))))) & ?[V_gi, V_vi]: (![V_x]: p(s(t_bool,h4s_extreals_extu_u_monou_u_increasing(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_gi))),s(TV_u_27a,V_x)))))) & (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_gi))),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),h4s_measures_fnu_u_minus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x)))) & (![V_i]: p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),V_vi),s(t_h4s_nums_num,V_i))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_gi),s(t_h4s_nums_num,V_i)))))))) & (![V_i, V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_gi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_minus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(TV_u_27a,V_x)))))) & (![V_i, V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_gi),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_x)))))) & s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_minus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_extreals_extreal)),V_gi))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ))))))))))))))).
fof(ah4s_lebesgues_integrableu_u_notu_u_infty, axiom, ![TV_u_27a]: ![V_m, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_lebesgues_integrable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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)))) & ![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))))) => ?[V_g]: (p(s(t_bool,h4s_lebesgues_integrable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g)))) & (![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x)))))) & (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => ~ (s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf))) & s(t_h4s_extreals_extreal,h4s_lebesgues_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))) = s(t_h4s_extreals_extreal,h4s_lebesgues_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g)))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_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_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_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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_measures_spaceu_u_def, axiom, ![TV_u_27a]: ![V_y, V_x]: s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y))))) = s(t_fun(TV_u_27a,t_bool),V_x)).
fof(ah4s_measures_subsetsu_u_def, axiom, ![TV_u_27a]: ![V_y, V_x]: s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_y)).
fof(ah4s_measures_finiteu_u_additivityu_u_sufficientu_u_foru_u_finiteu_u_spaces, axiom, ![TV_u_27a]: ![V_s, V_m]: ((p(s(t_bool,h4s_measures_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_s)))) & (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_s)))))) & (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_s))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_m)))))))) & p(s(t_bool,h4s_measures_additive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_s))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_m))))))))))) => p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_s))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_m)))))))))).
fof(ah4s_measures_MEASUREu_u_SPACEu_u_REDUCE, axiom, ![TV_u_27a]: ![V_m]: s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))) = s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_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_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_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_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_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_measures_FNu_u_PLUSu_u_POS, axiom, ![TV_u_27a]: ![V_x, V_g]: 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),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))),s(TV_u_27a,V_x))))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,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_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_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_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_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_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_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_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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,happ(s(t_fun(TV_u_27a,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(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,happ(s(t_fun(TV_u_27a,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,happ(s(t_fun(TV_u_27a,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_combins_Iu_u_ou_u_IDu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_combins_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_lebesgues_integralu_u_sequence, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_m, V_f, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_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_i))) = s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),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_i))))) => ![V_m, V_f]: ((![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))) & (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_posu_u_fnu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sup(s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_univ)))))))).
fof(ah4s_lebesgues_lemmau_u_fnu_u_inu_u_psfis, axiom, ![TV_u_27a]: ![V_n, V_m, V_f]: ((![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_x)))))) & (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))) => p(s(t_bool,h4s_bools_in(s(t_h4s_extreals_extreal,h4s_lebesgues_fnu_u_sequ_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_extreals_extreal,t_bool),h4s_lebesgues_psfis(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_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)))))))))).
fof(ah4s_measures_INu_u_MEASURABLEu_u_BORELu_u_FNu_u_MINUS, axiom, ![TV_u_27a]: ![V_f, V_a]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_minus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))).
fof(ah4s_measures_INu_u_MEASURABLEu_u_BORELu_u_FNu_u_PLUS, axiom, ![TV_u_27a]: ![V_f, V_a]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))))).
fof(ah4s_lebesgues_lemmau_u_fnu_u_sup, 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,happ(s(t_fun(TV_u_27a,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)))))).
fof(ah4s_measures_FNu_u_MINUSu_u_POS, axiom, ![TV_u_27a]: ![V_x, V_g]: 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),h4s_measures_fnu_u_minus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))),s(TV_u_27a,V_x))))))).
fof(ah4s_measures_indicatoru_u_fnu_u_def, axiom, ![TV_u_27a]: ![V_s, V_x]: s(t_h4s_extreals_extreal,h4s_measures_indicatoru_u_fn(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_measures_measureu_u_spaceu_u_def, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) <=> (p(s(t_bool,h4s_measures_sigmau_u_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) & (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & p(s(t_bool,h4s_measures_countablyu_u_additive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
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_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
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_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_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_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_measures_nullu_u_setu_u_def, axiom, ![TV_u_27a]: ![V_s, V_m]: (p(s(t_bool,h4s_measures_nullu_u_set(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_bool),V_s)))) <=> (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
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_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_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_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_extreals_subu_u_refl, 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))) => s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sub(s(t_h4s_extreals_extreal,V_x),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))))).
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_extreals_mulu_u_rone, axiom, ![V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) = s(t_h4s_extreals_extreal,V_x)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_extreals_leu_u_trans, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_y)))) & p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_y),s(t_h4s_extreals_extreal,V_z))))) => p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_z)))))).
fof(ah4s_bools_CONJu_u_SYM, 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))))).
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_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_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_measures_MEASUREu_u_SPACEu_u_UNION, axiom, ![TV_u_27a]: ![V_t, V_s, V_m]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))).
fof(ah4s_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_numu_u_notu_u_inftyu_c0, axiom, ![V_n]: ~ (s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = s(t_h4s_extreals_extreal,h4s_extreals_neginf))).
fof(ah4s_extreals_extrealu_u_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_leu_u_refl, axiom, ![V_x]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_x))))).
fof(ah4s_extreals_mulu_u_rzero, axiom, ![V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_extreals_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_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_extreals_numu_u_notu_u_inftyu_c1, 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_posinf))).
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_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_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_subu_u_defu_c1, axiom, ![V_a]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sub(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_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_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_subu_u_defu_c6, axiom, ![V_v3]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sub(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_posinf)).
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_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_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_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_measures_MEASUREu_u_SPACEu_u_EMPTYu_u_MEASURABLE, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))))).
fof(ah4s_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_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_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_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_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_reals_REALu_u_EQu_u_SUBu_u_LADD, axiom, ![V_z, V_y, V_x]: (s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))) <=> s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z))) = 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_reals_REALu_u_ADDu_u_COMM, axiom, ![V_y, 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_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_predu_u_sets_UNIONu_u_OVERu_u_INTER, axiom, ![TV_u_27a]: ![V_u, V_t, V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_u))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_u)))))).
fof(ah4s_extreals_extrealu_u_distinctu_c0, axiom, ~ (s(t_h4s_extreals_extreal,h4s_extreals_neginf) = s(t_h4s_extreals_extreal,h4s_extreals_posinf))).
fof(ah4s_predu_u_sets_INu_u_DIFF, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t)))))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
fof(ah4s_predu_u_sets_INTERu_u_COMM, axiom, ![TV_u_27a]: ![V_t, V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_s)))).
fof(ah4s_predu_u_sets_INTERu_u_EMPTYu_c0, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_INu_u_INTER, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_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_extreals_extrealu_u_distinctu_c1, axiom, ![V_a]: ~ (s(t_h4s_extreals_extreal,h4s_extreals_neginf) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_a))))).
fof(ah4s_predu_u_sets_INTERu_u_EMPTYu_c1, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_INu_u_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_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_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_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_combins_Su_u_DEF, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_x0, V_x1]: s(TV_u_27c,h4s_combins_s(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1)))))).
fof(ah4s_combins_Cu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_x, V_x0, V_x1]: s(TV_u_27c,h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27b,V_x0),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,V_x0)))).
fof(ah4s_markers_Abbrevu_u_def, axiom, ![V_x]: s(t_bool,h4s_markers_abbrev(s(t_bool,V_x))) = s(t_bool,V_x)).
fof(ah4s_extreals_extrealu_u_distinctu_c2, axiom, ![V_a]: ~ (s(t_h4s_extreals_extreal,h4s_extreals_posinf) = s(t_h4s_extreals_extreal,h4s_extreals_normal(s(t_h4s_realaxs_real,V_a))))).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => 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_dcu_u_cond, axiom, ![V_s, V_r, V_q, V_p]: (s(t_bool,V_p) = s(t_bool,h4s_bools_cond(s(t_bool,V_q),s(t_bool,V_r),s(t_bool,V_s))) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_s))))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_s))))) & ((~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_s)) | ~ (p(s(t_bool,V_p))))))))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
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(ch4s_lebesgues_finiteu_u_spaceu_u_integralu_u_reduce, conjecture, ![TV_u_27a]: ![V_m, V_f]: ((p(s(t_bool,h4s_measures_measureu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_fun(TV_u_27a,t_h4s_extreals_extreal),t_bool),h4s_measures_measurable(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))),s(t_h4s_pairs_prod(t_fun(t_h4s_extreals_extreal,t_bool),t_fun(t_fun(t_h4s_extreals_extreal,t_bool),t_bool)),h4s_measures_borel)))))) & (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => (~ (s(t_h4s_extreals_extreal,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_neginf)) & ~ (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_posinf)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_measures_mu_u_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))))))))) => s(t_h4s_extreals_extreal,h4s_lebesgues_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))) = s(t_h4s_extreals_extreal,h4s_lebesgues_finiteu_u_spaceu_u_integral(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))).
