%   ORIGINAL: 'h4/thm/pred_set/SUM_IMAGE_MONO_LESS_EQ_'
% 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 ==> (!x. 'h4/const/bool/IN' x s ==> 'h4/const/arithmetic/<=' (f x) (g x)) ==> 'h4/const/arithmetic/<=' ('h4/const/pred_set/SUM_IMAGE' f s) ('h4/const/pred_set/SUM_IMAGE' g 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: !g f s. 'h4/const/pred_set/FINITE' s ==> (!x. 'h4/const/bool/IN' x s ==> 'h4/const/arithmetic/<=' (happ f x) (happ g x)) ==> 'h4/const/arithmetic/<=' ('h4/const/pred_set/SUM_IMAGE' f s) ('h4/const/pred_set/SUM_IMAGE' 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_3f22448,V_3f22444]: ![F, G]: (![X]: s(V_3f22444,happ(s(fun(V_3f22448,V_3f22444),F),s(V_3f22448,X))) = s(V_3f22444,happ(s(fun(V_3f22448,V_3f22444),G),s(V_3f22448,X))) => s(fun(V_3f22448,V_3f22444),F) = s(fun(V_3f22448,V_3f22444),G))).
fof('h4/thm/pred_set/SUM_IMAGE_MONO_LESS_EQ_', conjecture, ![A]: ![G, F, S]: (p(s(bool,'h4/const/pred_set/FINITE'(s(fun(A,bool),S)))) => (![X]: (p(s(bool,'h4/const/bool/IN'(s(A,X),s(fun(A,bool),S)))) => p(s(bool,'h4/const/arithmetic/<='(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))))))) => p(s(bool,'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'),G),s(fun(A,bool),S))))))))).
