%   ORIGINAL: h4/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
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/lim/differentiable0: !x f. h4/lim/differentiable f x <=> (?l. h4/lim/diffl f l x)
% Assm: h4/poly/POLY__DIFF: !x l. h4/lim/diffl (\x0. h4/poly/poly l x0) (h4/poly/poly (h4/poly/diff l) x) x
% Goal: !x l. h4/lim/differentiable (\x0. h4/poly/poly l x0) x
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_lims_differentiable0]: !x f. h4/lim/differentiable f x <=> (?l. h4/lim/diffl f l x)
% Assm [h4s_polys_POLYu_u_DIFF]: !_0. (!l x0. happ (happ _0 l) x0 = h4/poly/poly l x0) ==> (!x l. h4/lim/diffl (happ _0 l) (h4/poly/poly (h4/poly/diff l) x) x)
% Goal: !_0. (!l x0. happ (happ _0 l) x0 = h4/poly/poly l x0) ==> (!x l. h4/lim/differentiable (happ _0 l) x)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q214219,TV_Q214215]: ![V_f, V_g]: (![V_x]: s(TV_Q214215,happ(s(t_fun(TV_Q214219,TV_Q214215),V_f),s(TV_Q214219,V_x))) = s(TV_Q214215,happ(s(t_fun(TV_Q214219,TV_Q214215),V_g),s(TV_Q214219,V_x))) => s(t_fun(TV_Q214219,TV_Q214215),V_f) = s(t_fun(TV_Q214219,TV_Q214215),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_lims_differentiable0, axiom, ![V_x, V_f]: (p(s(t_bool,h4s_lims_differentiable(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x)))) <=> ?[V_l]: p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_polys_POLYu_u_DIFF, axiom, ![V_uu_0]: (![V_l, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_lists_list(t_h4s_realaxs_real),V_l))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_l),s(t_h4s_realaxs_real,V_x0))) => ![V_x, V_l]: p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_lists_list(t_h4s_realaxs_real),V_l))),s(t_h4s_realaxs_real,h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_diff(s(t_h4s_lists_list(t_h4s_realaxs_real),V_l))),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_x)))))).
fof(ch4s_polys_POLYu_u_DIFFERENTIABLE, conjecture, ![V_uu_0]: (![V_l, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_lists_list(t_h4s_realaxs_real),V_l))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_l),s(t_h4s_realaxs_real,V_x0))) => ![V_x, V_l]: p(s(t_bool,h4s_lims_differentiable(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_lists_list(t_h4s_realaxs_real),V_l))),s(t_h4s_realaxs_real,V_x)))))).
