%   ORIGINAL: h4/lebesgue/integrable__fn__minus
% 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/FALSITY: !t. F ==> 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/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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/extreal/num__not__infty_c1: !n. ~(h4/extreal/extreal__of__num n = h4/extreal/PosInf)
% 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/FN__PLUS__POS__ID: !g. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (g x)) ==> h4/measure/fn__plus g = g
% Assm: h4/measure/FN__MINUS__POS__ZERO: !g. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (g x)) ==> h4/measure/fn__minus g = (\x. h4/extreal/extreal__of__num h4/num/0)
% 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/lebesgue/integrable__def: !m f. h4/lebesgue/integrable m f <=> 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 (h4/measure/fn__plus f) = h4/extreal/PosInf) /\ ~(h4/lebesgue/pos__fn__integral m (h4/measure/fn__minus f) = h4/extreal/PosInf)
% Assm: h4/lebesgue/pos__fn__integral__zero: !m. h4/measure/measure__space m ==> h4/lebesgue/pos__fn__integral m (\x. h4/extreal/extreal__of__num h4/num/0) = h4/extreal/extreal__of__num h4/num/0
% Goal: !m f. h4/measure/measure__space m /\ h4/lebesgue/integrable m f ==> h4/lebesgue/integrable m (h4/measure/fn__minus 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_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_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_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% 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_extreals_numu_u_notu_u_inftyu_c1]: !n. ~(h4/extreal/extreal__of__num n = h4/extreal/PosInf)
% 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_FNu_u_PLUSu_u_POSu_u_ID]: !g. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ g x)) ==> h4/measure/fn__plus g = g
% Assm [h4s_measures_FNu_u_MINUSu_u_POSu_u_ZERO]: !g. (!x. h4/extreal/extreal__le (h4/extreal/extreal__of__num h4/num/0) (happ g x)) ==> (!x. happ (h4/measure/fn__minus g) x = h4/extreal/extreal__of__num h4/num/0)
% 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_lebesgues_integrableu_u_def]: !m f. h4/lebesgue/integrable m f <=> 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 (h4/measure/fn__plus f) = h4/extreal/PosInf) /\ ~(h4/lebesgue/pos__fn__integral m (h4/measure/fn__minus f) = h4/extreal/PosInf)
% Assm [h4s_lebesgues_posu_u_fnu_u_integralu_u_zero]: !_0. (!x. happ _0 x = h4/extreal/extreal__of__num h4/num/0) ==> (!m. h4/measure/measure__space m ==> h4/lebesgue/pos__fn__integral m _0 = h4/extreal/extreal__of__num h4/num/0)
% Goal: !m f. h4/measure/measure__space m /\ h4/lebesgue/integrable m f ==> h4/lebesgue/integrable m (h4/measure/fn__minus 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_Q268701,TV_Q268697]: ![V_f, V_g]: (![V_x]: s(TV_Q268697,happ(s(t_fun(TV_Q268701,TV_Q268697),V_f),s(TV_Q268701,V_x))) = s(TV_Q268697,happ(s(t_fun(TV_Q268701,TV_Q268697),V_g),s(TV_Q268701,V_x))) => s(t_fun(TV_Q268701,TV_Q268697),V_f) = s(t_fun(TV_Q268701,TV_Q268697),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_FALSITY, axiom, ![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_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_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_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_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_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_FNu_u_PLUSu_u_POSu_u_ID, axiom, ![TV_u_27a]: ![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)))))) => 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(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_g))).
fof(ah4s_measures_FNu_u_MINUSu_u_POSu_u_ZERO, axiom, ![TV_u_27a]: ![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]: 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))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))).
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_lebesgues_integrableu_u_def, axiom, ![TV_u_27a]: ![V_m, V_f]: (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)))) <=> (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),h4s_measures_fnu_u_plus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f))))) = s(t_h4s_extreals_extreal,h4s_extreals_posinf)) & ~ (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_posinf)))))).
fof(ah4s_lebesgues_posu_u_fnu_u_integralu_u_zero, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_h4s_extreals_extreal,happ(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_uu_0),s(TV_u_27a,V_x))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) => ![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)))) => 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_uu_0))) = s(t_h4s_extreals_extreal,h4s_extreals_extrealu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ch4s_lebesgues_integrableu_u_fnu_u_minus, 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_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))))) => 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),h4s_measures_fnu_u_minus(s(t_fun(TV_u_27a,t_h4s_extreals_extreal),V_f)))))))).
