%   ORIGINAL: 'h4/thm/seq/SER_0_'
% 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 n. (!m. 'h4/const/arithmetic/<=' n m ==> f m = 'h4/const/real/real_of_num' 'h4/const/num/0') ==> 'h4/const/seq/sums' f ('h4/const/real/sum' ('h4/const/pair/,' 'h4/const/num/0' n) f)
%   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 n. (!m. 'h4/const/arithmetic/<=' n m ==> happ f m = 'h4/const/real/real_of_num' 'h4/const/num/0') ==> 'h4/const/seq/sums' f ('h4/const/real/sum' ('h4/const/pair/,' 'h4/const/num/0' n) 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_3f65286,V_3f65282]: ![F, G]: (![X]: s(V_3f65282,happ(s(fun(V_3f65286,V_3f65282),F),s(V_3f65286,X))) = s(V_3f65282,happ(s(fun(V_3f65286,V_3f65282),G),s(V_3f65286,X))) => s(fun(V_3f65286,V_3f65282),F) = s(fun(V_3f65286,V_3f65282),G))).
fof('h4/thm/seq/SER_0_', conjecture, ![F, N]: (![M]: (p(s(bool,'h4/const/arithmetic/<='(s('h4/type/num/num',N),s('h4/type/num/num',M)))) => s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),F),s('h4/type/num/num',M))) = s('h4/type/realax/real','h4/const/real/real_of_num'(s('h4/type/num/num','h4/const/num/0')))) => p(s(bool,'h4/const/seq/sums'(s(fun('h4/type/num/num','h4/type/realax/real'),F),s('h4/type/realax/real','h4/const/real/sum'(s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num','h4/const/num/0'),s('h4/type/num/num',N))),s(fun('h4/type/num/num','h4/type/realax/real'),F)))))))).
