%   ORIGINAL: 'h4/thm/real/SUM_SUB_'
% 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 g m n. 'h4/const/real/sum' ('h4/const/pair/,' m n) (\n0. 'h4/const/real/real_sub' (f n0) (g n0)) = 'h4/const/real/real_sub' ('h4/const/real/sum' ('h4/const/pair/,' m n) f) ('h4/const/real/sum' ('h4/const/pair/,' m n) g)
%   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 g n0. happ (happ (happ _0 f) g) n0 = 'h4/const/real/real_sub' (happ f n0) (happ g n0)) ==> (!f g m n. 'h4/const/real/sum' ('h4/const/pair/,' m n) (happ (happ _0 f) g) = 'h4/const/real/real_sub' ('h4/const/real/sum' ('h4/const/pair/,' m n) f) ('h4/const/real/sum' ('h4/const/pair/,' m n) g))
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_3f94419,V_3f94415]: ![F, G]: (![X]: s(V_3f94415,happ(s(fun(V_3f94419,V_3f94415),F),s(V_3f94419,X))) = s(V_3f94415,happ(s(fun(V_3f94419,V_3f94415),G),s(V_3f94419,X))) => s(fun(V_3f94419,V_3f94415),F) = s(fun(V_3f94419,V_3f94415),G))).
fof('h4/thm/real/SUM_SUB_', conjecture, ![V__0]: (![F, G, N0]: 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(fun('h4/type/num/num','h4/type/realax/real'),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(fun('h4/type/num/num','h4/type/realax/real'),G))),s('h4/type/num/num',N0))) = s('h4/type/realax/real','h4/const/real/real_sub'(s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),F),s('h4/type/num/num',N0))),s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),G),s('h4/type/num/num',N0))))) => ![F, G, M, 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',M),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(fun('h4/type/num/num','h4/type/realax/real'),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(fun('h4/type/num/num','h4/type/realax/real'),G))))) = s('h4/type/realax/real','h4/const/real/real_sub'(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',M),s('h4/type/num/num',N))),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',M),s('h4/type/num/num',N))),s(fun('h4/type/num/num','h4/type/realax/real'),G))))))).
