%   ORIGINAL: 'h4/thm/integral/SUM_DIFFS_'
% 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: !d m n. 'h4/const/real/sum' ('h4/const/pair/,' m n) (\i. 'h4/const/real/real_sub' (d ('h4/const/num/SUC' i)) (d i)) = 'h4/const/real/real_sub' (d ('h4/const/arithmetic/+' m n)) (d m)
%   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. (!d i. happ (happ _0 d) i = 'h4/const/real/real_sub' (happ d ('h4/const/num/SUC' i)) (happ d i)) ==> (!d m n. 'h4/const/real/sum' ('h4/const/pair/,' m n) (happ _0 d) = 'h4/const/real/real_sub' (happ d ('h4/const/arithmetic/+' m n)) (happ d m))
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_3f40734,V_3f40730]: ![F, G]: (![X]: s(V_3f40730,happ(s(fun(V_3f40734,V_3f40730),F),s(V_3f40734,X))) = s(V_3f40730,happ(s(fun(V_3f40734,V_3f40730),G),s(V_3f40734,X))) => s(fun(V_3f40734,V_3f40730),F) = s(fun(V_3f40734,V_3f40730),G))).
fof('h4/thm/integral/SUM_DIFFS_', conjecture, ![V__0]: (![D, I]: 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'),D))),s('h4/type/num/num',I))) = 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'),D),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',I))))),s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),D),s('h4/type/num/num',I))))) => ![D, 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')),V__0),s(fun('h4/type/num/num','h4/type/realax/real'),D))))) = 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'),D),s('h4/type/num/num','h4/const/arithmetic/+'(s('h4/type/num/num',M),s('h4/type/num/num',N))))),s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),D),s('h4/type/num/num',M))))))).
