%   ORIGINAL: 'h4/thm/seq/SER_ABS_'
% 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. 'h4/const/seq/summable' (\n. 'h4/const/real/abs' (f n)) ==> 'h4/const/real/real_lte' ('h4/const/real/abs' ('h4/const/seq/suminf' f)) ('h4/const/seq/suminf' (\n. 'h4/const/real/abs' (f n)))
%   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 n. happ (happ _0 f) n = 'h4/const/real/abs' (happ f n)) ==> (!f. 'h4/const/seq/summable' (happ _0 f) ==> 'h4/const/real/real_lte' ('h4/const/real/abs' ('h4/const/seq/suminf' f)) ('h4/const/seq/suminf' (happ _0 f)))
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_3f65761,V_3f65757]: ![F, G]: (![X]: s(V_3f65757,happ(s(fun(V_3f65761,V_3f65757),F),s(V_3f65761,X))) = s(V_3f65757,happ(s(fun(V_3f65761,V_3f65757),G),s(V_3f65761,X))) => s(fun(V_3f65761,V_3f65757),F) = s(fun(V_3f65761,V_3f65757),G))).
fof('h4/thm/seq/SER_ABS_', conjecture, ![V__0]: (![F, N]: s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),happ(s(fun(fun('h4/type/num/num','h4/type/realax/real'),fun('h4/type/num/num','h4/type/realax/real')),V__0),s(fun('h4/type/num/num','h4/type/realax/real'),F))),s('h4/type/num/num',N))) = s('h4/type/realax/real','h4/const/real/abs'(s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),F),s('h4/type/num/num',N))))) => ![F]: (p(s(bool,'h4/const/seq/summable'(s(fun('h4/type/num/num','h4/type/realax/real'),happ(s(fun(fun('h4/type/num/num','h4/type/realax/real'),fun('h4/type/num/num','h4/type/realax/real')),V__0),s(fun('h4/type/num/num','h4/type/realax/real'),F)))))) => p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real','h4/const/real/abs'(s('h4/type/realax/real','h4/const/seq/suminf'(s(fun('h4/type/num/num','h4/type/realax/real'),F))))),s('h4/type/realax/real','h4/const/seq/suminf'(s(fun('h4/type/num/num','h4/type/realax/real'),happ(s(fun(fun('h4/type/num/num','h4/type/realax/real'),fun('h4/type/num/num','h4/type/realax/real')),V__0),s(fun('h4/type/num/num','h4/type/realax/real'),F))))))))))).
