%   ORIGINAL: 'h4/thm/pred_set/PROD_IMAGE_THM_e1'
% 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
% Goal: !f e s. 'h4/const/pred_set/FINITE' s ==> 'h4/const/pred_set/PROD_IMAGE' f ('h4/const/pred_set/INSERT' e s) = 'h4/const/arithmetic/*' (f e) ('h4/const/pred_set/PROD_IMAGE' f ('h4/const/pred_set/DELETE' s e))
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !f e s. 'h4/const/pred_set/FINITE' s ==> 'h4/const/pred_set/PROD_IMAGE' f ('h4/const/pred_set/INSERT' e s) = 'h4/const/arithmetic/*' (happ f e) ('h4/const/pred_set/PROD_IMAGE' f ('h4/const/pred_set/DELETE' s e))
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f22723,V_3f22719]: ![F, G]: (![X]: s(V_3f22719,happ(s(fun(V_3f22723,V_3f22719),F),s(V_3f22723,X))) = s(V_3f22719,happ(s(fun(V_3f22723,V_3f22719),G),s(V_3f22723,X))) => s(fun(V_3f22723,V_3f22719),F) = s(fun(V_3f22723,V_3f22719),G))).
fof('h4/thm/pred_set/PROD_IMAGE_THM_e1', conjecture, ![A]: ![F, E, S]: (p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S)))) => s('h4/type/num/num','h4/const/pred_set/PROD_IMAGE'(s(fun(A,'h4/type/num/num'),F),s(fun(A,bool),'h4/const/pred_set/INSERT'(s(A,E),s(fun(A,bool),S))))) = s('h4/type/num/num','h4/const/arithmetic/*'(s('h4/type/num/num',happ(s(fun(A,'h4/type/num/num'),F),s(A,E))),s('h4/type/num/num','h4/const/pred_set/PROD_IMAGE'(s(fun(A,'h4/type/num/num'),F),s(fun(A,bool),'h4/const/pred_set/DELETE'(s(fun(A,bool),S),s(A,E))))))))).
