%   ORIGINAL: h4/pred__set/ABS__DIFF__SUM__IMAGE
% 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/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/ABS__DIFF__ZERO_c1: !n. h4/arithmetic/ABS__DIFF h4/num/0 n = n
% Assm: h4/arithmetic/ABS__DIFF__SUMS: !n2 n1 m2 m1. h4/arithmetic/_3C_3D (h4/arithmetic/ABS__DIFF (h4/arithmetic/_2B n1 n2) (h4/arithmetic/_2B m1 m2)) (h4/arithmetic/_2B (h4/arithmetic/ABS__DIFF n1 m1) (h4/arithmetic/ABS__DIFF n2 m2))
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/pred__set/DELETE__NON__ELEMENT: !x s. ~h4/bool/IN x s <=> h4/pred__set/DELETE s x = s
% Assm: h4/pred__set/FINITE__INDUCT: !P. P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ P s ==> (!e. ~h4/bool/IN e s ==> P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> P s)
% Assm: h4/pred__set/SUM__IMAGE__THM_c0: !f. h4/pred__set/SUM__IMAGE f h4/pred__set/EMPTY = h4/num/0
% Assm: h4/pred__set/SUM__IMAGE__THM_c1: !s f e. h4/pred__set/FINITE s ==> h4/pred__set/SUM__IMAGE f (h4/pred__set/INSERT e s) = h4/arithmetic/_2B (f e) (h4/pred__set/SUM__IMAGE f (h4/pred__set/DELETE s e))
% Goal: !s g f. h4/pred__set/FINITE s ==> h4/arithmetic/_3C_3D (h4/arithmetic/ABS__DIFF (h4/pred__set/SUM__IMAGE f s) (h4/pred__set/SUM__IMAGE g s)) (h4/pred__set/SUM__IMAGE (\x. h4/arithmetic/ABS__DIFF (f x) (g x)) s)
%   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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_ABSu_u_DIFFu_u_ZEROu_c1]: !n. h4/arithmetic/ABS__DIFF h4/num/0 n = n
% Assm [h4s_arithmetics_ABSu_u_DIFFu_u_SUMS]: !n2 n1 m2 m1. h4/arithmetic/_3C_3D (h4/arithmetic/ABS__DIFF (h4/arithmetic/_2B n1 n2) (h4/arithmetic/_2B m1 m2)) (h4/arithmetic/_2B (h4/arithmetic/ABS__DIFF n1 m1) (h4/arithmetic/ABS__DIFF n2 m2))
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_predu_u_sets_DELETEu_u_NONu_u_ELEMENT]: !x s. ~h4/bool/IN x s <=> h4/pred__set/DELETE s x = s
% Assm [h4s_predu_u_sets_FINITEu_u_INDUCT]: !P. happ P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ happ P s ==> (!e. ~h4/bool/IN e s ==> happ P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> happ P s)
% Assm [h4s_predu_u_sets_SUMu_u_IMAGEu_u_THMu_c0]: !f. h4/pred__set/SUM__IMAGE f h4/pred__set/EMPTY = h4/num/0
% Assm [h4s_predu_u_sets_SUMu_u_IMAGEu_u_THMu_c1]: !s f e. h4/pred__set/FINITE s ==> h4/pred__set/SUM__IMAGE f (h4/pred__set/INSERT e s) = h4/arithmetic/_2B (happ f e) (h4/pred__set/SUM__IMAGE f (h4/pred__set/DELETE s e))
% Goal: !_0. (!f g x. happ (happ (happ _0 f) g) x = h4/arithmetic/ABS__DIFF (happ f x) (happ g x)) ==> (!s g f. h4/pred__set/FINITE s ==> h4/arithmetic/_3C_3D (h4/arithmetic/ABS__DIFF (h4/pred__set/SUM__IMAGE f s) (h4/pred__set/SUM__IMAGE g s)) (h4/pred__set/SUM__IMAGE (happ (happ _0 f) g) s))
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_Q154408,TV_Q154404]: ![V_f, V_g]: (![V_x]: s(TV_Q154404,happ(s(t_fun(TV_Q154408,TV_Q154404),V_f),s(TV_Q154408,V_x))) = s(TV_Q154404,happ(s(t_fun(TV_Q154408,TV_Q154404),V_g),s(TV_Q154408,V_x))) => s(t_fun(TV_Q154408,TV_Q154404),V_f) = s(t_fun(TV_Q154408,TV_Q154404),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,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_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> s(t_bool,V_t) = s(t_bool,f0))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(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_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_NOTu_u_LEQ, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(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_nums_num,V_n)))).
fof(ah4s_arithmetics_ABSu_u_DIFFu_u_ZEROu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_absu_u_diff(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_ABSu_u_DIFFu_u_SUMS, axiom, ![V_n2, V_n1, V_m2, V_m1]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_absu_u_diff(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n1),s(t_h4s_nums_num,V_n2))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m1),s(t_h4s_nums_num,V_m2))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_absu_u_diff(s(t_h4s_nums_num,V_n1),s(t_h4s_nums_num,V_m1))),s(t_h4s_nums_num,h4s_arithmetics_absu_u_diff(s(t_h4s_nums_num,V_n2),s(t_h4s_nums_num,V_m2))))))))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f0)).
fof(ah4s_predu_u_sets_DELETEu_u_NONu_u_ELEMENT, axiom, ![TV_u_27a]: ![V_x, V_s]: (~ (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_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),V_s))).
fof(ah4s_predu_u_sets_FINITEu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & ![V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_e]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))))))) => ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_SUMu_u_IMAGEu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_f]: s(t_h4s_nums_num,h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_predu_u_sets_SUMu_u_IMAGEu_u_THMu_c1, axiom, ![TV_u_27a]: ![V_s, V_f, V_e]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_nums_num,h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_e))),s(t_h4s_nums_num,h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_e))))))))).
fof(ch4s_predu_u_sets_ABSu_u_DIFFu_u_SUMu_u_IMAGE, conjecture, ![TV_u_27a]: ![V_uu_0]: (![V_f, V_g, V_x]: s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_h4s_nums_num)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_h4s_nums_num))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_g))),s(TV_u_27a,V_x))) = s(t_h4s_nums_num,h4s_arithmetics_absu_u_diff(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_x))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_g),s(TV_u_27a,V_x))))) => ![V_s, V_g, V_f]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_absu_u_diff(s(t_h4s_nums_num,h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_g),s(t_fun(TV_u_27a,t_bool),V_s))))),s(t_h4s_nums_num,h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_h4s_nums_num)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_h4s_nums_num))),V_uu_0),s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_g))),s(t_fun(TV_u_27a,t_bool),V_s))))))))).
