%   ORIGINAL: 'h4/thm/poly/POLY_DIFFERENTIABLE_'
% 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: !l x. 'h4/const/lim/differentiable' (\x0. 'h4/const/poly/poly' l x0) x
%   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. (!l x0. happ (happ _0 l) x0 = 'h4/const/poly/poly' l x0) ==> (!l x. 'h4/const/lim/differentiable' (happ _0 l) x)
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_3f46484,V_3f46480]: ![F, G]: (![X]: s(V_3f46480,happ(s(fun(V_3f46484,V_3f46480),F),s(V_3f46484,X))) = s(V_3f46480,happ(s(fun(V_3f46484,V_3f46480),G),s(V_3f46484,X))) => s(fun(V_3f46484,V_3f46480),F) = s(fun(V_3f46484,V_3f46480),G))).
fof('h4/thm/poly/POLY_DIFFERENTIABLE_', conjecture, ![V__0]: (![L, X0]: s('h4/type/realax/real',happ(s(fun('h4/type/realax/real','h4/type/realax/real'),happ(s(fun('h4/type/list/list'('h4/type/realax/real'),fun('h4/type/realax/real','h4/type/realax/real')),V__0),s('h4/type/list/list'('h4/type/realax/real'),L))),s('h4/type/realax/real',X0))) = s('h4/type/realax/real','h4/const/poly/poly'(s('h4/type/list/list'('h4/type/realax/real'),L),s('h4/type/realax/real',X0))) => ![L, X]: p(s(bool,'h4/const/lim/differentiable'(s(fun('h4/type/realax/real','h4/type/realax/real'),happ(s(fun('h4/type/list/list'('h4/type/realax/real'),fun('h4/type/realax/real','h4/type/realax/real')),V__0),s('h4/type/list/list'('h4/type/realax/real'),L))),s('h4/type/realax/real',X)))))).
