%   ORIGINAL: h4/pred__set/SUM__SET__THM_c1
% 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__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_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/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/arithmetic/EQ__MONO__ADD__EQ: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% 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))
% Assm: h4/pred__set/SUM__SET__DEF: h4/pred__set/SUM__SET = h4/pred__set/SUM__IMAGE h4/combin/I
% Goal: !x s. h4/pred__set/FINITE s ==> h4/pred__set/SUM__SET (h4/pred__set/INSERT x s) = h4/arithmetic/_2B x (h4/pred__set/SUM__SET (h4/pred__set/DELETE s x))
%   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_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_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_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ]: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm [h4s_predu_u_sets_SUMu_u_IMAGEu_u_THMu_c1]: !s f e. h4/pred__set/FINITE s ==> happ (h4/pred__set/SUM__IMAGE f) (h4/pred__set/INSERT e s) = h4/arithmetic/_2B (happ f e) (happ (h4/pred__set/SUM__IMAGE f) (h4/pred__set/DELETE s e))
% Assm [h4s_predu_u_sets_SUMu_u_SETu_u_DEF]: h4/pred__set/SUM__SET = h4/pred__set/SUM__IMAGE h4/combin/I
% Goal: !x s. h4/pred__set/FINITE s ==> happ h4/pred__set/SUM__SET (h4/pred__set/INSERT x s) = h4/arithmetic/_2B x (happ h4/pred__set/SUM__SET (h4/pred__set/DELETE s x))
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_Q154508,TV_Q154504]: ![V_f, V_g]: (![V_x]: s(TV_Q154504,happ(s(t_fun(TV_Q154508,TV_Q154504),V_f),s(TV_Q154508,V_x))) = s(TV_Q154504,happ(s(t_fun(TV_Q154508,TV_Q154504),V_g),s(TV_Q154508,V_x))) => s(t_fun(TV_Q154508,TV_Q154504),V_f) = s(t_fun(TV_Q154508,TV_Q154504),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, V_n, V_m]: (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_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
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,happ(s(t_fun(t_fun(TV_u_27a,t_bool),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,happ(s(t_fun(t_fun(TV_u_27a,t_bool),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(ah4s_predu_u_sets_SUMu_u_SETu_u_DEF, axiom, s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set) = s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_combins_i)))).
fof(ch4s_predu_u_sets_SUMu_u_SETu_u_THMu_c1, conjecture, ![V_x, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,V_x),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_delete(s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_h4s_nums_num,V_x))))))))).
