%   ORIGINAL: h4/measure/LAMBDA__SYSTEM__EMPTY
% 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/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> 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/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/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% 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/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% 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/INTER__EMPTY_c0: !s. h4/pred__set/INTER h4/pred__set/EMPTY s = h4/pred__set/EMPTY
% Assm: h4/pred__set/DIFF__EMPTY: !s. h4/pred__set/DIFF s h4/pred__set/EMPTY = s
% Assm: h4/real/REAL__ADD__LID: !x. h4/realax/real__add (h4/real/real__of__num h4/num/0) x = x
% Assm: h4/measure/algebra__def: !a. h4/measure/algebra a <=> h4/measure/subset__class (h4/measure/space a) (h4/measure/subsets a) /\ h4/bool/IN h4/pred__set/EMPTY (h4/measure/subsets a) /\ (!s. h4/bool/IN s (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/DIFF (h4/measure/space a) s) (h4/measure/subsets a)) /\ (!s t. h4/bool/IN s (h4/measure/subsets a) /\ h4/bool/IN t (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/UNION s t) (h4/measure/subsets a))
% Assm: h4/measure/measure__def: !sts sp mu. h4/measure/measure (h4/pair/_2C sp (h4/pair/_2C sts mu)) = mu
% Assm: h4/measure/positive__def: !m. h4/measure/positive m <=> h4/measure/measure m h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0 /\ (!s. h4/bool/IN s (h4/measure/measurable__sets m) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/measure/measure m s))
% Assm: h4/measure/lambda__system__def: !lam gen. h4/measure/lambda__system gen lam = h4/pred__set/GSPEC (\l. h4/pair/_2C l (h4/bool/IN l (h4/measure/subsets gen) /\ (!s. h4/bool/IN s (h4/measure/subsets gen) ==> h4/realax/real__add (lam (h4/pred__set/INTER l s)) (lam (h4/pred__set/INTER (h4/pred__set/DIFF (h4/measure/space gen) l) s)) = lam s)))
% Assm: h4/measure/ALGEBRA__INTER__SPACE: !s a. h4/measure/algebra a /\ h4/bool/IN s (h4/measure/subsets a) ==> h4/pred__set/INTER (h4/measure/space a) s = s /\ h4/pred__set/INTER s (h4/measure/space a) = s
% Goal: !lam g0. h4/measure/algebra g0 /\ h4/measure/positive (h4/pair/_2C (h4/measure/space g0) (h4/pair/_2C (h4/measure/subsets g0) lam)) ==> h4/bool/IN h4/pred__set/EMPTY (h4/measure/lambda__system g0 lam)
%   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_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> 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_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_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% 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_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% 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_INTERu_u_EMPTYu_c0]: !s. h4/pred__set/INTER h4/pred__set/EMPTY s = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_DIFFu_u_EMPTY]: !s. h4/pred__set/DIFF s h4/pred__set/EMPTY = s
% Assm [h4s_reals_REALu_u_ADDu_u_LID]: !x. h4/realax/real__add (h4/real/real__of__num h4/num/0) x = x
% Assm [h4s_measures_algebrau_u_def]: !a. h4/measure/algebra a <=> h4/measure/subset__class (h4/measure/space a) (h4/measure/subsets a) /\ h4/bool/IN h4/pred__set/EMPTY (h4/measure/subsets a) /\ (!s. h4/bool/IN s (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/DIFF (h4/measure/space a) s) (h4/measure/subsets a)) /\ (!s t. h4/bool/IN s (h4/measure/subsets a) /\ h4/bool/IN t (h4/measure/subsets a) ==> h4/bool/IN (h4/pred__set/UNION s t) (h4/measure/subsets a))
% Assm [h4s_measures_measureu_u_def]: !sts sp mu. h4/measure/measure (h4/pair/_2C sp (h4/pair/_2C sts mu)) = mu
% Assm [h4s_measures_positiveu_u_def]: !m. h4/measure/positive m <=> happ (h4/measure/measure m) h4/pred__set/EMPTY = h4/real/real__of__num h4/num/0 /\ (!s. h4/bool/IN s (h4/measure/measurable__sets m) ==> h4/real/real__lte (h4/real/real__of__num h4/num/0) (happ (h4/measure/measure m) s))
% Assm [h4s_measures_lambdau_u_systemu_u_def]: !_0. (!gen lam l. ?v. (v <=> h4/bool/IN l (h4/measure/subsets gen) /\ (!s. h4/bool/IN s (h4/measure/subsets gen) ==> h4/realax/real__add (happ lam (h4/pred__set/INTER l s)) (happ lam (h4/pred__set/INTER (h4/pred__set/DIFF (h4/measure/space gen) l) s)) = happ lam s)) /\ happ (happ (happ _0 gen) lam) l = h4/pair/_2C l v) ==> (!lam gen. h4/measure/lambda__system gen lam = h4/pred__set/GSPEC (happ (happ _0 gen) lam))
% Assm [h4s_measures_ALGEBRAu_u_INTERu_u_SPACE]: !s a. h4/measure/algebra a /\ h4/bool/IN s (h4/measure/subsets a) ==> h4/pred__set/INTER (h4/measure/space a) s = s /\ h4/pred__set/INTER s (h4/measure/space a) = s
% Goal: !lam g0. h4/measure/algebra g0 /\ h4/measure/positive (h4/pair/_2C (h4/measure/space g0) (h4/pair/_2C (h4/measure/subsets g0) lam)) ==> h4/bool/IN h4/pred__set/EMPTY (h4/measure/lambda__system g0 lam)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q110500,TV_Q110496]: ![V_f, V_g]: (![V_x]: s(TV_Q110496,happ(s(t_fun(TV_Q110500,TV_Q110496),V_f),s(TV_Q110500,V_x))) = s(TV_Q110496,happ(s(t_fun(TV_Q110500,TV_Q110496),V_g),s(TV_Q110500,V_x))) => s(t_fun(TV_Q110500,TV_Q110496),V_f) = s(t_fun(TV_Q110500,TV_Q110496),V_g))).
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_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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_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_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_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_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),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(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_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
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_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_DIFFu_u_EMPTY, axiom, ![TV_u_27a]: ![V_s]: 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),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),V_s)).
fof(ah4s_reals_REALu_u_ADDu_u_LID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_measures_algebrau_u_def, axiom, ![TV_u_27a]: ![V_a]: (p(s(t_bool,h4s_measures_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))) <=> (p(s(t_bool,h4s_measures_subsetu_u_class(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))) & (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))) & (![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))))))) & ![V_s, V_t]: ((p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))))))) => p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_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_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))))))))))).
fof(ah4s_measures_measureu_u_def, axiom, ![TV_u_27a]: ![V_sts, V_sp, V_mu]: s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_sp),s(t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real)),h4s_pairs_u_2c(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_sts),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_mu))))))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_mu)).
fof(ah4s_measures_positiveu_u_def, axiom, ![TV_u_27a]: ![V_m]: (p(s(t_bool,h4s_measures_positive(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))) <=> (s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & ![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_measurableu_u_sets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m)))))) => p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),h4s_measures_measure(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real))),V_m))),s(t_fun(TV_u_27a,t_bool),V_s)))))))))).
fof(ah4s_measures_lambdau_u_systemu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_gen, V_lam, V_l]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_l),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_gen)))))) & ![V_s]: (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_gen)))))) => s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_lam),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_l),s(t_fun(TV_u_27a,t_bool),V_s))))),s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_lam),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_diff(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_gen))),s(t_fun(TV_u_27a,t_bool),V_l))),s(t_fun(TV_u_27a,t_bool),V_s))))))) = s(t_h4s_realaxs_real,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_lam),s(t_fun(TV_u_27a,t_bool),V_s)))))) & s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool)))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_gen))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_lam))),s(t_fun(TV_u_27a,t_bool),V_l))) = s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_l),s(t_bool,V_v)))) => ![V_lam, V_gen]: s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_lambdau_u_system(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_gen),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_lam))) = s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool)))),V_uu_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_gen))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_lam))))))).
fof(ah4s_measures_ALGEBRAu_u_INTERu_u_SPACE, axiom, ![TV_u_27a]: ![V_s, V_a]: ((p(s(t_bool,h4s_measures_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a)))) & p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_measures_subsets(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))))))) => (s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),h4s_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),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_measures_space(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_a))))) = s(t_fun(TV_u_27a,t_bool),V_s)))).
fof(ch4s_measures_LAMBDAu_u_SYSTEMu_u_EMPTY, conjecture, ![TV_u_27a]: ![V_lam, V_g0]: ((p(s(t_bool,h4s_measures_algebra(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_g0)))) & 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_g0))),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_g0))),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_lam))))))))) => 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_lambdau_u_system(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),V_g0),s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_realaxs_real),V_lam)))))))).
