%   ORIGINAL: h4/lebesgue/pos__simple__fn__not__infty
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/WF__EMPTY__REL: h4/relation/WF h4/relation/EMPTY__REL
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/real/REAL__MUL__RZERO: !x. h4/realax/real__mul x (h4/real/real__of__num h4/num/0) = h4/real/real__of__num h4/num/0
% Assm: h4/real/eq__ints_c0: !n m. h4/real/real__of__num n = h4/real/real__of__num m <=> n = m
% Assm: h4/real/lt__int_c0: !n m. h4/realax/real__lt (h4/real/real__of__num n) (h4/real/real__of__num m) <=> h4/prim__rec/_3C n m
% Assm: h4/extreal/extreal__repfns_c0: !a. h4/extreal/_20_40ind__typeextreal3 (h4/extreal/_20_40ind__typeextreal4 a) = a
% Assm: h4/extreal/extreal__repfns_c1: !r. (\a0. !_27extreal_27. (!a00. a00 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ a00 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a. a00 = (\a1. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) a1 (\n. h4/ind__type/BOTTOM)) a) ==> _27extreal_27 a00) ==> _27extreal_27 a0) r <=> h4/extreal/_20_40ind__typeextreal4 (h4/extreal/_20_40ind__typeextreal3 r) = r
% Assm: h4/extreal/hidden____20__40ind____typeextreal0____def: h4/extreal/_20_40ind__typeextreal0 = h4/extreal/_20_40ind__typeextreal3 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/extreal/hidden____20__40ind____typeextreal1____def: h4/extreal/_20_40ind__typeextreal1 = h4/extreal/_20_40ind__typeextreal3 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/extreal/hidden____20__40ind____typeextreal2____def: h4/extreal/_20_40ind__typeextreal2 = (\a. h4/extreal/_20_40ind__typeextreal3 ((\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) a0 (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/extreal/NegInf0: h4/extreal/NegInf = h4/extreal/_20_40ind__typeextreal0
% Assm: h4/extreal/PosInf0: h4/extreal/PosInf = h4/extreal/_20_40ind__typeextreal1
% Assm: h4/extreal/Normal0: h4/extreal/Normal = h4/extreal/_20_40ind__typeextreal2
% Assm: h4/extreal/extreal__case__def_c0: !v1 v f. h4/extreal/extreal__CASE h4/extreal/NegInf v v1 f = v
% Assm: h4/extreal/extreal__case__def_c1: !v1 v f. h4/extreal/extreal__CASE h4/extreal/PosInf v v1 f = v1
% Assm: h4/extreal/extreal__case__def_c2: !v1 v f a. h4/extreal/extreal__CASE (h4/extreal/Normal a) v v1 f = f a
% Assm: h4/extreal/extreal__distinct_c1: !a. ~(h4/extreal/NegInf = h4/extreal/Normal a)
% Assm: h4/extreal/extreal__distinct_c2: !a. ~(h4/extreal/PosInf = h4/extreal/Normal a)
% Assm: h4/extreal/extreal__of__num__def: !n. h4/extreal/extreal__of__num n = h4/extreal/Normal (h4/real/real__of__num n)
% Assm: h4/extreal/extreal__mul__tupled__primitive__def: h4/extreal/extreal__mul__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R)) (\extreal__mul__tupled a. h4/pair/pair__CASE a (\v v1. h4/extreal/extreal__CASE v (h4/extreal/extreal__CASE v1 (h4/combin/I h4/extreal/PosInf) (h4/combin/I h4/extreal/NegInf) (\y. h4/combin/I (h4/bool/COND (y = h4/real/real__of__num h4/num/0) (h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) y) h4/extreal/NegInf h4/extreal/PosInf)))) (h4/extreal/extreal__CASE v1 (h4/combin/I h4/extreal/NegInf) (h4/combin/I h4/extreal/PosInf) (\y_27. h4/combin/I (h4/bool/COND (y_27 = h4/real/real__of__num h4/num/0) (h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) y_27) h4/extreal/PosInf h4/extreal/NegInf)))) (\x. h4/extreal/extreal__CASE v1 (h4/combin/I (h4/bool/COND (x = h4/real/real__of__num h4/num/0) (h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) x) h4/extreal/NegInf h4/extreal/PosInf))) (h4/combin/I (h4/bool/COND (x = h4/real/real__of__num h4/num/0) (h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) x) h4/extreal/PosInf h4/extreal/NegInf))) (\y_27_27. h4/combin/I (h4/extreal/Normal (h4/realax/real__mul x y_27_27))))))
% Assm: h4/extreal/extreal__mul__curried__def: !x1 x. h4/extreal/extreal__mul x x1 = h4/extreal/extreal__mul__tupled (h4/pair/_2C x x1)
% Assm: h4/extreal/mul__rone: !x. h4/extreal/extreal__mul x (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm: h4/extreal/EXTREAL__SUM__IMAGE__NOT__INFTY_c0: !s f. h4/pred__set/FINITE s /\ (!x. h4/bool/IN x s ==> ~(f x = h4/extreal/NegInf)) ==> ~(h4/extreal/EXTREAL__SUM__IMAGE f s = h4/extreal/NegInf)
% Assm: h4/extreal/EXTREAL__SUM__IMAGE__NOT__INFTY_c1: !s f. h4/pred__set/FINITE s /\ (!x. h4/bool/IN x s ==> ~(f x = h4/extreal/PosInf)) ==> ~(h4/extreal/EXTREAL__SUM__IMAGE f s = h4/extreal/PosInf)
% 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/pos__simple__fn__def: !x s m f a. h4/measure/pos__simple__fn m f s a x <=> (!t. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (f t)) /\ (!t. h4/bool/IN t (h4/measure/m__space m) ==> f t = h4/extreal/EXTREAL__SUM__IMAGE (\i. h4/extreal/extreal__mul (h4/extreal/Normal (x i)) (h4/measure/indicator__fn (a i) t)) s) /\ (!i. h4/bool/IN i s ==> h4/bool/IN (a i) (h4/measure/measurable__sets m)) /\ h4/pred__set/FINITE s /\ (!i. h4/bool/IN i s ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (x i)) /\ (!i j. h4/bool/IN i s /\ h4/bool/IN j s /\ ~(i = j) ==> h4/pred__set/DISJOINT (a i) (a j)) /\ h4/pred__set/BIGUNION (h4/pred__set/IMAGE a s) = h4/measure/m__space m
% Goal: !x s m f a. h4/measure/pos__simple__fn m f s a x ==> (!x0. h4/bool/IN x0 (h4/measure/m__space m) ==> ~(f x0 = h4/extreal/NegInf) /\ ~(f x0 = h4/extreal/PosInf))
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_WFu_u_EMPTYu_u_REL]: h4/relation/WF h4/relation/EMPTY__REL
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = happ f n
% Assm [h4s_reals_REALu_u_MULu_u_RZERO]: !x. h4/realax/real__mul x (h4/real/real__of__num h4/num/0) = h4/real/real__of__num h4/num/0
% Assm [h4s_reals_equ_u_intsu_c0]: !n m. h4/real/real__of__num n = h4/real/real__of__num m <=> n = m
% Assm [h4s_reals_ltu_u_intu_c0]: !n m. h4/realax/real__lt (h4/real/real__of__num n) (h4/real/real__of__num m) <=> h4/prim__rec/_3C n m
% Assm [h4s_extreals_extrealu_u_repfnsu_c0]: !a. h4/extreal/_20_40ind__typeextreal3 (h4/extreal/_20_40ind__typeextreal4 a) = a
% Assm [h4s_extreals_extrealu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27extreal_27. (!a00. a00 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ a00 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB _0 \/ (?a. a00 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) a _0) ==> happ _27extreal_27 a00) ==> happ _27extreal_27 r) <=> h4/extreal/_20_40ind__typeextreal4 (h4/extreal/_20_40ind__typeextreal3 r) = r)
% Assm [h4s_extreals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeextreal0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/extreal/_20_40ind__typeextreal0 = h4/extreal/_20_40ind__typeextreal3 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_extreals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeextreal1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/extreal/_20_40ind__typeextreal1 = h4/extreal/_20_40ind__typeextreal3 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB _0)
% Assm [h4s_extreals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeextreal2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/extreal/_20_40ind__typeextreal2 x = h4/extreal/_20_40ind__typeextreal3 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) x _0))
% Assm [h4s_extreals_NegInf0]: h4/extreal/NegInf = h4/extreal/_20_40ind__typeextreal0
% Assm [h4s_extreals_PosInf0]: h4/extreal/PosInf = h4/extreal/_20_40ind__typeextreal1
% Assm [h4s_extreals_Normal0]: h4/extreal/Normal = h4/extreal/_20_40ind__typeextreal2
% Assm [h4s_extreals_extrealu_u_caseu_u_defu_c0]: !v1 v f. h4/extreal/extreal__CASE h4/extreal/NegInf v v1 f = v
% Assm [h4s_extreals_extrealu_u_caseu_u_defu_c1]: !v1 v f. h4/extreal/extreal__CASE h4/extreal/PosInf v v1 f = v1
% Assm [h4s_extreals_extrealu_u_caseu_u_defu_c2]: !v1 v f a. h4/extreal/extreal__CASE (happ h4/extreal/Normal a) v v1 f = happ f a
% Assm [h4s_extreals_extrealu_u_distinctu_c1]: !a. ~(h4/extreal/NegInf = happ h4/extreal/Normal a)
% Assm [h4s_extreals_extrealu_u_distinctu_c2]: !a. ~(h4/extreal/PosInf = happ h4/extreal/Normal a)
% Assm [h4s_extreals_extrealu_u_ofu_u_numu_u_def]: !n. h4/extreal/extreal__of__num n = happ h4/extreal/Normal (h4/real/real__of__num n)
% Assm [h4s_extreals_extrealu_u_mulu_u_tupledu_u_primitiveu_u_def]: !_8. (!x y_27_27. happ (happ _8 x) y_27_27 = h4/combin/I (happ h4/extreal/Normal (h4/realax/real__mul x y_27_27))) ==> (!_7. (!v1 x. ?v. (v <=> x = h4/real/real__of__num h4/num/0) /\ (?v'. (v' <=> x = h4/real/real__of__num h4/num/0) /\ happ (happ _7 v1) x = h4/extreal/extreal__CASE v1 (h4/combin/I (h4/bool/COND v' (happ h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) x) h4/extreal/NegInf h4/extreal/PosInf))) (h4/combin/I (h4/bool/COND v (happ h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) x) h4/extreal/PosInf h4/extreal/NegInf))) (happ _8 x))) ==> (!_6. (!y_27. ?v. (v <=> y_27 = h4/real/real__of__num h4/num/0) /\ happ _6 y_27 = h4/combin/I (h4/bool/COND v (happ h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) y_27) h4/extreal/PosInf h4/extreal/NegInf))) ==> (!_5. (!y. ?v. (v <=> y = h4/real/real__of__num h4/num/0) /\ happ _5 y = h4/combin/I (h4/bool/COND v (happ h4/extreal/Normal (h4/real/real__of__num h4/num/0)) (h4/bool/COND (h4/realax/real__lt (h4/real/real__of__num h4/num/0) y) h4/extreal/NegInf h4/extreal/PosInf))) ==> (!_4. (!v v1. happ (happ _4 v) v1 = h4/extreal/extreal__CASE v (h4/extreal/extreal__CASE v1 (h4/combin/I h4/extreal/PosInf) (h4/combin/I h4/extreal/NegInf) _5) (h4/extreal/extreal__CASE v1 (h4/combin/I h4/extreal/NegInf) (h4/combin/I h4/extreal/PosInf) _6) (happ _7 v1)) ==> (!_3. (!v. happ _3 v = happ _4 v) ==> (!_2. (!a. happ _2 a = h4/pair/pair__CASE a _3) ==> (!_1. (!extreal__mul__tupled. happ _1 extreal__mul__tupled = _2) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R) ==> h4/extreal/extreal__mul__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))
% Assm [h4s_extreals_extrealu_u_mulu_u_curriedu_u_def]: !x1 x. h4/extreal/extreal__mul x x1 = happ h4/extreal/extreal__mul__tupled (h4/pair/_2C x x1)
% Assm [h4s_extreals_mulu_u_rone]: !x. h4/extreal/extreal__mul x (h4/extreal/extreal__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm [h4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_NOTu_u_INFTYu_c0]: !s f. h4/pred__set/FINITE s /\ (!x. h4/bool/IN x s ==> ~(happ f x = h4/extreal/NegInf)) ==> ~(h4/extreal/EXTREAL__SUM__IMAGE f s = h4/extreal/NegInf)
% Assm [h4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_NOTu_u_INFTYu_c1]: !s f. h4/pred__set/FINITE s /\ (!x. h4/bool/IN x s ==> ~(happ f x = h4/extreal/PosInf)) ==> ~(h4/extreal/EXTREAL__SUM__IMAGE f s = h4/extreal/PosInf)
% 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_posu_u_simpleu_u_fnu_u_def]: !_0. (!x a t i. happ (happ (happ (happ _0 x) a) t) i = h4/extreal/extreal__mul (happ h4/extreal/Normal (happ x i)) (h4/measure/indicator__fn (happ a i) t)) ==> (!x s m f a. h4/measure/pos__simple__fn m f s a x <=> (!t. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ f t)) /\ (!t. h4/bool/IN t (h4/measure/m__space m) ==> happ f t = h4/extreal/EXTREAL__SUM__IMAGE (happ (happ (happ _0 x) a) t) s) /\ (!i. h4/bool/IN i s ==> h4/bool/IN (happ a i) (h4/measure/measurable__sets m)) /\ h4/pred__set/FINITE s /\ (!i. h4/bool/IN i s ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ x i)) /\ (!i j. h4/bool/IN i s /\ h4/bool/IN j s /\ ~(i = j) ==> h4/pred__set/DISJOINT (happ a i) (happ a j)) /\ h4/pred__set/BIGUNION (h4/pred__set/IMAGE a s) = h4/measure/m__space m)
% Goal: !x s m f a. h4/measure/pos__simple__fn m f s a x ==> (!x0. h4/bool/IN x0 (h4/measure/m__space m) ==> ~(happ f x0 = h4/extreal/NegInf) /\ ~(happ f x0 = h4/extreal/PosInf))
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_Q266974,TV_Q266970]: ![V_f, V_g]: (![V_x]: s(TV_Q266970,happ(s(t_fun(TV_Q266974,TV_Q266970),V_f),s(TV_Q266974,V_x))) = s(TV_Q266970,happ(s(t_fun(TV_Q266974,TV_Q266970),V_g),s(TV_Q266974,V_x))) => s(t_fun(TV_Q266974,TV_Q266970),V_f) = s(t_fun(TV_Q266974,TV_Q266970),V_g))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => 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,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_relations_WFu_u_EMPTYu_u_REL, axiom, ![TV_u_27a]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel))))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![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),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_primu_u_recs_NOTu_u_LESSu_u_0, axiom, ![V_n]: ~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_reals_REALu_u_MULu_u_RZERO, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_reals_equ_u_intsu_c0, axiom, ![V_n, V_m]: (s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,V_m))).
fof(ah4s_reals_ltu_u_intu_c0, axiom, ![V_n, V_m]: s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_extreals_extrealu_u_repfnsu_c0, axiom, ![V_a]: s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal3(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_extreals_u_20u_40indu_u_typeextreal4(s(t_h4s_extreals_extreal,V_a))))) = s(t_h4s_extreals_extreal,V_a)).
fof(ah4s_extreals_extrealu_u_repfnsu_c1, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27extrealu_27]: (![V_a00]: ((s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))) | (s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))) | ?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_realaxs_real,V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),t_bool),V_uu_27extrealu_27),s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),t_bool),V_uu_27extrealu_27),s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_extreals_u_20u_40indu_u_typeextreal4(s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal3(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_r)))).
fof(ah4s_extreals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeextreal0u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal0) = s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal3(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))))).
fof(ah4s_extreals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeextreal1u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal1) = s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal3(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))))).
fof(ah4s_extreals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeextreal2u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_u_20u_40indu_u_typeextreal2),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal3(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_realaxs_real,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))))).
fof(ah4s_extreals_NegInf0, axiom, s(t_h4s_extreals_extreal,h4s_extreals_neginf) = s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal0)).
fof(ah4s_extreals_PosInf0, axiom, s(t_h4s_extreals_extreal,h4s_extreals_posinf) = s(t_h4s_extreals_extreal,h4s_extreals_u_20u_40indu_u_typeextreal1)).
fof(ah4s_extreals_Normal0, axiom, s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal) = s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_u_20u_40indu_u_typeextreal2)).
fof(ah4s_extreals_extrealu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f]: s(TV_u_27a,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(TV_u_27a,V_v),s(TV_u_27a,V_v1),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f))) = s(TV_u_27a,V_v)).
fof(ah4s_extreals_extrealu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f]: s(TV_u_27a,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(TV_u_27a,V_v),s(TV_u_27a,V_v1),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f))) = s(TV_u_27a,V_v1)).
fof(ah4s_extreals_extrealu_u_caseu_u_defu_c2, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f, V_a]: s(TV_u_27a,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,V_a))),s(TV_u_27a,V_v),s(TV_u_27a,V_v1),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f))) = s(TV_u_27a,happ(s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f),s(t_h4s_realaxs_real,V_a)))).
fof(ah4s_extreals_extrealu_u_distinctu_c1, axiom, ![V_a]: ~ (s(t_h4s_extreals_extreal,h4s_extreals_neginf) = s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,V_a))))).
fof(ah4s_extreals_extrealu_u_distinctu_c2, axiom, ![V_a]: ~ (s(t_h4s_extreals_extreal,h4s_extreals_posinf) = s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,V_a))))).
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,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n)))))).
fof(ah4s_extreals_extrealu_u_mulu_u_tupledu_u_primitiveu_u_def, axiom, ![V_uu_8]: (![V_x, V_yu_27u_27]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal)),V_uu_8),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_yu_27u_27))) = s(t_h4s_extreals_extreal,h4s_combins_i(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_yu_27u_27))))))) => ![V_uu_7]: (![V_v1, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) & s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal)),V_uu_7),s(t_h4s_extreals_extreal,V_v1))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,V_v1),s(t_h4s_extreals_extreal,h4s_combins_i(s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,V_vi_),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))),s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,h4s_extreals_posinf))))))),s(t_h4s_extreals_extreal,h4s_combins_i(s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))),s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,h4s_extreals_neginf))))))),s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal)),V_uu_8),s(t_h4s_realaxs_real,V_x))))))) => ![V_uu_6]: (![V_yu_27]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_realaxs_real,V_yu_27) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) & s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),V_uu_6),s(t_h4s_realaxs_real,V_yu_27))) = s(t_h4s_extreals_extreal,h4s_combins_i(s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_yu_27))),s(t_h4s_extreals_extreal,h4s_extreals_posinf),s(t_h4s_extreals_extreal,h4s_extreals_neginf)))))))) => ![V_uu_5]: (![V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_realaxs_real,V_y) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) & s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),V_uu_5),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_extreals_extreal,h4s_combins_i(s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_extreals_extreal,h4s_bools_cond(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_y))),s(t_h4s_extreals_extreal,h4s_extreals_neginf),s(t_h4s_extreals_extreal,h4s_extreals_posinf)))))))) => ![V_uu_4]: (![V_v, V_v1]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal)),V_uu_4),s(t_h4s_extreals_extreal,V_v))),s(t_h4s_extreals_extreal,V_v1))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,V_v),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,V_v1),s(t_h4s_extreals_extreal,h4s_combins_i(s(t_h4s_extreals_extreal,h4s_extreals_posinf))),s(t_h4s_extreals_extreal,h4s_combins_i(s(t_h4s_extreals_extreal,h4s_extreals_neginf))),s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),V_uu_5))),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_case(s(t_h4s_extreals_extreal,V_v1),s(t_h4s_extreals_extreal,h4s_combins_i(s(t_h4s_extreals_extreal,h4s_extreals_neginf))),s(t_h4s_extreals_extreal,h4s_combins_i(s(t_h4s_extreals_extreal,h4s_extreals_posinf))),s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),V_uu_6))),s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal)),V_uu_7),s(t_h4s_extreals_extreal,V_v1))))) => ![V_uu_3]: (![V_v]: s(t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal)),V_uu_3),s(t_h4s_extreals_extreal,V_v))) = s(t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal),happ(s(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal)),V_uu_4),s(t_h4s_extreals_extreal,V_v))) => ![V_uu_2]: (![V_a]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_h4s_extreals_extreal),V_uu_2),s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),V_a))) = s(t_h4s_extreals_extreal,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),V_a),s(t_fun(t_h4s_extreals_extreal,t_fun(t_h4s_extreals_extreal,t_h4s_extreals_extreal)),V_uu_3))) => ![V_uu_1]: (![V_extrealu_u_mulu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_h4s_extreals_extreal),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_h4s_extreals_extreal),t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_h4s_extreals_extreal)),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_h4s_extreals_extreal),V_extrealu_u_mulu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_h4s_extreals_extreal),V_uu_2) => ![V_uu_0]: (![V_R]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_bool)),V_R))) = s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_bool)),V_R))) => s(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_h4s_extreals_extreal),h4s_extreals_extrealu_u_mulu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_h4s_extreals_extreal),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_h4s_extreals_extreal),t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_h4s_extreals_extreal)),V_uu_1))))))))))))).
fof(ah4s_extreals_extrealu_u_mulu_u_curriedu_u_def, axiom, ![V_x1, V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_x1))) = s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),t_h4s_extreals_extreal),h4s_extreals_extrealu_u_mulu_u_tupled),s(t_h4s_pairs_prod(t_h4s_extreals_extreal,t_h4s_extreals_extreal),h4s_pairs_u_2c(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,V_x1)))))).
fof(ah4s_extreals_mulu_u_rone, axiom, ![V_x]: s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,V_x),s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) = s(t_h4s_extreals_extreal,V_x)).
fof(ah4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_NOTu_u_INFTYu_c0, axiom, ![TV_u_27a]: ![V_s, V_f]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => ~ (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,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_extreals_extreal,h4s_extreals_neginf)))).
fof(ah4s_extreals_EXTREALu_u_SUMu_u_IMAGEu_u_NOTu_u_INFTYu_c1, axiom, ![TV_u_27a]: ![V_s, V_f]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => ~ (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)))) => ~ (s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)))).
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_posu_u_simpleu_u_fnu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_a, V_t, V_i]: s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_nums_num,t_h4s_extreals_extreal),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a))),s(TV_u_27a,V_t))),s(t_h4s_nums_num,V_i))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_mul(s(t_h4s_extreals_extreal,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_extreals_extreal),h4s_extreals_normal),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_nums_num,V_i))))),s(t_h4s_extreals_extreal,h4s_measures_indicatoru_u_fn(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_h4s_nums_num,V_i))),s(TV_u_27a,V_t))))) => ![V_x, V_s, V_m, V_f, V_a]: (p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x)))) <=> (![V_t]: p(s(t_bool,h4s_extreals_extrealu_u_le(s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(TV_u_27a,V_t)))))) & (![V_t]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_t),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_t))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_sumu_u_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_bool)),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_h4s_realaxs_real),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_h4s_extreals_extreal)))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a))),s(TV_u_27a,V_t))),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) & (![V_i]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_h4s_nums_num,V_i))),s(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(t_h4s_nums_num,t_bool),V_s)))) & (![V_i]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x),s(t_h4s_nums_num,V_i))))))) & (![V_i, V_j]: ((p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) & (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_j),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) & ~ (s(t_h4s_nums_num,V_i) = s(t_h4s_nums_num,V_j)))) => p(s(t_bool,h4s_predu_u_sets_disjoint(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_h4s_nums_num,V_i))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_h4s_nums_num,V_j))))))) & s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_bigunion(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = 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(ch4s_lebesgues_posu_u_simpleu_u_fnu_u_notu_u_infty, conjecture, ![TV_u_27a]: ![V_x, V_s, V_m, V_f, V_a]: (p(s(t_bool,h4s_measures_posu_u_simpleu_u_fn(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m),s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f),s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_x)))) => ![V_x0]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x0),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_x0))) = 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_x0))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)))))).
