%   ORIGINAL: 'h4/thm/pred_set/SUM_IMAGE_UNION_'
% 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 t. 'h4/const/pred_set/FINITE' s /\ 'h4/const/pred_set/FINITE' t ==> 'h4/const/pred_set/SUM_IMAGE' f ('h4/const/pred_set/UNION' s t) = 'h4/const/arithmetic/-' ('h4/const/arithmetic/+' ('h4/const/pred_set/SUM_IMAGE' f s) ('h4/const/pred_set/SUM_IMAGE' f t)) ('h4/const/pred_set/SUM_IMAGE' f ('h4/const/pred_set/INTER' s t))
%   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 t. 'h4/const/pred_set/FINITE' s /\ 'h4/const/pred_set/FINITE' t ==> 'h4/const/pred_set/SUM_IMAGE' f ('h4/const/pred_set/UNION' s t) = 'h4/const/arithmetic/-' ('h4/const/arithmetic/+' ('h4/const/pred_set/SUM_IMAGE' f s) ('h4/const/pred_set/SUM_IMAGE' f t)) ('h4/const/pred_set/SUM_IMAGE' f ('h4/const/pred_set/INTER' s t))
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_3f22273,V_3f22269]: ![F, G]: (![X]: s(V_3f22269,happ(s(fun(V_3f22273,V_3f22269),F),s(V_3f22273,X))) = s(V_3f22269,happ(s(fun(V_3f22273,V_3f22269),G),s(V_3f22273,X))) => s(fun(V_3f22273,V_3f22269),F) = s(fun(V_3f22273,V_3f22269),G))).
fof('h4/thm/pred_set/SUM_IMAGE_UNION_', conjecture, ![A]: ![F, S, T]: ((p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S)))) & p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),T))))) => 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/UNION'(s(fun(A,bool),S),s(fun(A,bool),T))))) = s('h4/type/num/num','h4/const/arithmetic/-'(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','h4/const/pred_set/SUM_IMAGE'(s(fun(A,'h4/type/num/num'),F),s(fun(A,bool),T))))),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/INTER'(s(fun(A,bool),S),s(fun(A,bool),T))))))))).
