%   ORIGINAL: h4/real__sigma/REAL__SUM__IMAGE__EQ
% 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/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/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/real__sigma/REAL__SUM__IMAGE__IN__IF: !P. h4/pred__set/FINITE P ==> (!f. h4/real__sigma/REAL__SUM__IMAGE f P = h4/real__sigma/REAL__SUM__IMAGE (\x. h4/bool/COND (h4/bool/IN x P) (f x) (h4/real/real__of__num h4/num/0)) P)
% Goal: !s f_27 f. h4/pred__set/FINITE s /\ (!x. h4/bool/IN x s ==> f x = f_27 x) ==> h4/real__sigma/REAL__SUM__IMAGE f s = h4/real__sigma/REAL__SUM__IMAGE f_27 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_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_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_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_INu_u_IF]: !_0. (!P f x. happ (happ (happ _0 P) f) x = h4/bool/COND (h4/bool/IN x P) (happ f x) (h4/real/real__of__num h4/num/0)) ==> (!P. h4/pred__set/FINITE P ==> (!f. h4/real__sigma/REAL__SUM__IMAGE f P = h4/real__sigma/REAL__SUM__IMAGE (happ (happ _0 P) f) P))
% Goal: !s f_27 f. h4/pred__set/FINITE s /\ (!x. h4/bool/IN x s ==> happ f x = happ f_27 x) ==> h4/real__sigma/REAL__SUM__IMAGE f s = h4/real__sigma/REAL__SUM__IMAGE f_27 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_Q278265,TV_Q278261]: ![V_f, V_g]: (![V_x]: s(TV_Q278261,happ(s(t_fun(TV_Q278265,TV_Q278261),V_f),s(TV_Q278265,V_x))) = s(TV_Q278261,happ(s(t_fun(TV_Q278265,TV_Q278261),V_g),s(TV_Q278265,V_x))) => s(t_fun(TV_Q278265,TV_Q278261),V_f) = s(t_fun(TV_Q278265,TV_Q278261),V_g))).
fof(ah4s_bools_TRUTH, axiom, 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_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_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_INu_u_IF, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_f, V_x]: s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f))),s(TV_u_27a,V_x))) = s(t_h4s_realaxs_real,h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(TV_u_27a,V_x))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) => ![V_P]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_P)))) => ![V_f]: s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_h4s_realaxs_real),t_fun(TV_u_27a,t_h4s_realaxs_real))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f))),s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ch4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_EQ, conjecture, ![TV_u_27a]: ![V_s, V_fu_27, 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_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(TV_u_27a,V_x))) = s(t_h4s_realaxs_real,happ(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_fu_27),s(TV_u_27a,V_x))))) => s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_f),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),V_fu_27),s(t_fun(TV_u_27a,t_bool),V_s))))).
