%   ORIGINAL: 'h4/thm/seq/SER_PAIR_'
% 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' f ==> 'h4/const/seq/sums' (\n. 'h4/const/real/sum' ('h4/const/pair/,' ('h4/const/arithmetic/*' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 'h4/const/arithmetic/ZERO')) n) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 'h4/const/arithmetic/ZERO'))) f) ('h4/const/seq/suminf' 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: !_0. (!f n. happ (happ _0 f) n = 'h4/const/real/sum' ('h4/const/pair/,' ('h4/const/arithmetic/*' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 'h4/const/arithmetic/ZERO')) n) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT2' 'h4/const/arithmetic/ZERO'))) f) ==> (!f. 'h4/const/seq/summable' f ==> 'h4/const/seq/sums' (happ _0 f) ('h4/const/seq/suminf' 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_3f65386,V_3f65382]: ![F, G]: (![X]: s(V_3f65382,happ(s(fun(V_3f65386,V_3f65382),F),s(V_3f65386,X))) = s(V_3f65382,happ(s(fun(V_3f65386,V_3f65382),G),s(V_3f65386,X))) => s(fun(V_3f65386,V_3f65382),F) = s(fun(V_3f65386,V_3f65382),G))).
fof('h4/thm/seq/SER_PAIR_', 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/sum'(s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num','h4/const/arithmetic/*'(s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num','h4/const/arithmetic/ZERO'))))),s('h4/type/num/num',N))),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num','h4/const/arithmetic/ZERO'))))))),s(fun('h4/type/num/num','h4/type/realax/real'),F))) => ![F]: (p(s(bool,'h4/const/seq/summable'(s(fun('h4/type/num/num','h4/type/realax/real'),F)))) => p(s(bool,'h4/const/seq/sums'(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/realax/real','h4/const/seq/suminf'(s(fun('h4/type/num/num','h4/type/realax/real'),F))))))))).
