%   ORIGINAL: 'h4/thm/pred_set/SUM_SAME_IMAGE_'
% 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: !P. 'h4/const/pred_set/FINITE' P ==> (!f p. 'h4/const/bool/IN' p P /\ (!q. 'h4/const/bool/IN' q P ==> f p = f q) ==> 'h4/const/pred_set/SUM_IMAGE' f P = 'h4/const/arithmetic/*' ('h4/const/pred_set/CARD' P) (f p))
%   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: !P. 'h4/const/pred_set/FINITE' P ==> (!f p. 'h4/const/bool/IN' p P /\ (!q. 'h4/const/bool/IN' q P ==> happ f p = happ f q) ==> 'h4/const/pred_set/SUM_IMAGE' f P = 'h4/const/arithmetic/*' ('h4/const/pred_set/CARD' P) (happ f p))
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_3f22348,V_3f22344]: ![F, G]: (![X]: s(V_3f22344,happ(s(fun(V_3f22348,V_3f22344),F),s(V_3f22348,X))) = s(V_3f22344,happ(s(fun(V_3f22348,V_3f22344),G),s(V_3f22348,X))) => s(fun(V_3f22348,V_3f22344),F) = s(fun(V_3f22348,V_3f22344),G))).
fof('h4/thm/pred_set/SUM_SAME_IMAGE_', conjecture, ![A]: ![P]: (p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),P)))) => ![F, P0]: ((p(s(bool,'h4/const/bool/IN'(s(A,P0),s(fun(A,bool),P)))) & ![Q]: (p(s(bool,'h4/const/bool/IN'(s(A,Q),s(fun(A,bool),P)))) => s('h4/type/num/num',happ(s(fun(A,'h4/type/num/num'),F),s(A,P0))) = s('h4/type/num/num',happ(s(fun(A,'h4/type/num/num'),F),s(A,Q))))) => s('h4/type/num/num','h4/const/pred_set/SUM_IMAGE'(s(fun(A,'h4/type/num/num'),F),s(fun(A,bool),P))) = s('h4/type/num/num','h4/const/arithmetic/*'(s('h4/type/num/num','h4/const/pred_set/CARD'(s(fun(A,bool),P))),s('h4/type/num/num',happ(s(fun(A,'h4/type/num/num'),F),s(A,P0)))))))).
