%   ORIGINAL: 'h4/thm/pred_set/SUM_IMAGE_DELETE_'
% 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 ==> (!e. 'h4/const/pred_set/SUM_IMAGE' f ('h4/const/pred_set/DELETE' s e) = 'h4/const/bool/COND' ('h4/const/bool/IN' e s) ('h4/const/arithmetic/-' ('h4/const/pred_set/SUM_IMAGE' f s) (f e)) ('h4/const/pred_set/SUM_IMAGE' f s))
%   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 ==> (!e. 'h4/const/pred_set/SUM_IMAGE' f ('h4/const/pred_set/DELETE' s e) = 'h4/const/bool/COND' ('h4/const/bool/IN' e s) ('h4/const/arithmetic/-' ('h4/const/pred_set/SUM_IMAGE' f s) (happ f e)) ('h4/const/pred_set/SUM_IMAGE' f s))
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_3f22248,V_3f22244]: ![F, G]: (![X]: s(V_3f22244,happ(s(fun(V_3f22248,V_3f22244),F),s(V_3f22248,X))) = s(V_3f22244,happ(s(fun(V_3f22248,V_3f22244),G),s(V_3f22248,X))) => s(fun(V_3f22248,V_3f22244),F) = s(fun(V_3f22248,V_3f22244),G))).
fof('h4/thm/pred_set/SUM_IMAGE_DELETE_', conjecture, ![A]: ![F, S]: (p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S)))) => ![E]: s('h4/type/num/num','h4/const/pred_set/SUM_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))))) = s('h4/type/num/num','h4/const/bool/COND'(s(bool,'h4/const/bool/IN'(s(A,E),s(fun(A,bool),S))),s('h4/type/num/num','h4/const/arithmetic/-'(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',happ(s(fun(A,'h4/type/num/num'),F),s(A,E))))),s('h4/type/num/num','h4/const/pred_set/SUM_IMAGE'(s(fun(A,'h4/type/num/num'),F),s(fun(A,bool),S))))))).
