%   ORIGINAL: 'h4/thm/pred_set/ABS_DIFF_SUM_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: !g f s. 'h4/const/pred_set/FINITE' s ==> 'h4/const/arithmetic/<=' ('h4/const/arithmetic/ABS_DIFF' ('h4/const/pred_set/SUM_IMAGE' f s) ('h4/const/pred_set/SUM_IMAGE' g s)) ('h4/const/pred_set/SUM_IMAGE' (\x. 'h4/const/arithmetic/ABS_DIFF' (f x) (g x)) 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: !_0. (!f g x. happ (happ (happ _0 f) g) x = 'h4/const/arithmetic/ABS_DIFF' (happ f x) (happ g x)) ==> (!g f s. 'h4/const/pred_set/FINITE' s ==> 'h4/const/arithmetic/<=' ('h4/const/arithmetic/ABS_DIFF' ('h4/const/pred_set/SUM_IMAGE' f s) ('h4/const/pred_set/SUM_IMAGE' g s)) ('h4/const/pred_set/SUM_IMAGE' (happ (happ _0 f) g) 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_3f22423,V_3f22419]: ![F, G]: (![X]: s(V_3f22419,happ(s(fun(V_3f22423,V_3f22419),F),s(V_3f22423,X))) = s(V_3f22419,happ(s(fun(V_3f22423,V_3f22419),G),s(V_3f22423,X))) => s(fun(V_3f22423,V_3f22419),F) = s(fun(V_3f22423,V_3f22419),G))).
fof('h4/thm/pred_set/ABS_DIFF_SUM_IMAGE_', conjecture, ![A]: ![V__0]: (![F, G, X]: s('h4/type/num/num',happ(s(fun(A,'h4/type/num/num'),happ(s(fun(fun(A,'h4/type/num/num'),fun(A,'h4/type/num/num')),happ(s(fun(fun(A,'h4/type/num/num'),fun(fun(A,'h4/type/num/num'),fun(A,'h4/type/num/num'))),V__0),s(fun(A,'h4/type/num/num'),F))),s(fun(A,'h4/type/num/num'),G))),s(A,X))) = s('h4/type/num/num','h4/const/arithmetic/ABS_DIFF'(s('h4/type/num/num',happ(s(fun(A,'h4/type/num/num'),F),s(A,X))),s('h4/type/num/num',happ(s(fun(A,'h4/type/num/num'),G),s(A,X))))) => ![G, F, S]: (p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S)))) => p(s(bool,'h4/const/arithmetic/<='(s('h4/type/num/num','h4/const/arithmetic/ABS_DIFF'(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'),G),s(fun(A,bool),S))))),s('h4/type/num/num','h4/const/pred_set/SUM_IMAGE'(s(fun(A,'h4/type/num/num'),happ(s(fun(fun(A,'h4/type/num/num'),fun(A,'h4/type/num/num')),happ(s(fun(fun(A,'h4/type/num/num'),fun(fun(A,'h4/type/num/num'),fun(A,'h4/type/num/num'))),V__0),s(fun(A,'h4/type/num/num'),F))),s(fun(A,'h4/type/num/num'),G))),s(fun(A,bool),S))))))))).
