%   ORIGINAL: 'h4/thm/real/SUM_GROUP_'
% 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: !n k f. 'h4/const/real/sum' ('h4/const/pair/,' 'h4/const/num/0' n) (\m. 'h4/const/real/sum' ('h4/const/pair/,' ('h4/const/arithmetic/*' m k) k) f) = 'h4/const/real/sum' ('h4/const/pair/,' 'h4/const/num/0' ('h4/const/arithmetic/*' n k)) 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. (!k f m. happ (happ (happ _0 k) f) m = 'h4/const/real/sum' ('h4/const/pair/,' ('h4/const/arithmetic/*' m k) k) f) ==> (!n k f. 'h4/const/real/sum' ('h4/const/pair/,' 'h4/const/num/0' n) (happ (happ _0 k) f) = 'h4/const/real/sum' ('h4/const/pair/,' 'h4/const/num/0' ('h4/const/arithmetic/*' n k)) 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_3f94519,V_3f94515]: ![F, G]: (![X]: s(V_3f94515,happ(s(fun(V_3f94519,V_3f94515),F),s(V_3f94519,X))) = s(V_3f94515,happ(s(fun(V_3f94519,V_3f94515),G),s(V_3f94519,X))) => s(fun(V_3f94519,V_3f94515),F) = s(fun(V_3f94519,V_3f94515),G))).
fof('h4/thm/real/SUM_GROUP_', conjecture, ![V__0]: (![K, F, M]: 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')),happ(s(fun('h4/type/num/num',fun(fun('h4/type/num/num','h4/type/realax/real'),fun('h4/type/num/num','h4/type/realax/real'))),V__0),s('h4/type/num/num',K))),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/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',M),s('h4/type/num/num',K))),s('h4/type/num/num',K))),s(fun('h4/type/num/num','h4/type/realax/real'),F))) => ![N, K, 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'),happ(s(fun(fun('h4/type/num/num','h4/type/realax/real'),fun('h4/type/num/num','h4/type/realax/real')),happ(s(fun('h4/type/num/num',fun(fun('h4/type/num/num','h4/type/realax/real'),fun('h4/type/num/num','h4/type/realax/real'))),V__0),s('h4/type/num/num',K))),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','h4/const/arithmetic/*'(s('h4/type/num/num',N),s('h4/type/num/num',K))))),s(fun('h4/type/num/num','h4/type/realax/real'),F))))).
