%   ORIGINAL: 'h4/thm/pred_set/SUM_IMAGE_ZERO_'
% 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 s. 'h4/const/pred_set/FINITE' s ==> ('h4/const/pred_set/SUM_IMAGE' f s = 'h4/const/num/0' <=> (!x. 'h4/const/bool/IN' x s ==> f x = 'h4/const/num/0'))
%   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 s. 'h4/const/pred_set/FINITE' s ==> ('h4/const/pred_set/SUM_IMAGE' f s = 'h4/const/num/0' <=> (!x. 'h4/const/bool/IN' x s ==> happ f x = 'h4/const/num/0'))
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_3f22398,V_3f22394]: ![F, G]: (![X]: s(V_3f22394,happ(s(fun(V_3f22398,V_3f22394),F),s(V_3f22398,X))) = s(V_3f22394,happ(s(fun(V_3f22398,V_3f22394),G),s(V_3f22398,X))) => s(fun(V_3f22398,V_3f22394),F) = s(fun(V_3f22398,V_3f22394),G))).
fof('h4/thm/pred_set/SUM_IMAGE_ZERO_', conjecture, ![A]: ![F, S]: (p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S)))) => (s('h4/type/num/num','h4/const/pred_set/SUM_IMAGE'(s(fun(A,'h4/type/num/num'),F),s(fun(A,bool),S))) = s('h4/type/num/num','h4/const/num/0') <=> ![X]: (p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),S)))) => s('h4/type/num/num',happ(s(fun(A,'h4/type/num/num'),F),s(A,X))) = s('h4/type/num/num','h4/const/num/0'))))).
