%   ORIGINAL: h4/poly/POLY__MVT
% 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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/lim/DIFF__UNIQ: !x m l f. h4/lim/diffl f l x /\ h4/lim/diffl f m x ==> l = m
% Assm: h4/lim/MVT: !f b a. h4/realax/real__lt a b /\ (!x. h4/real/real__lte a x /\ h4/real/real__lte x b ==> h4/lim/contl f x) /\ (!x. h4/realax/real__lt a x /\ h4/realax/real__lt x b ==> h4/lim/differentiable f x) ==> (?l z. h4/realax/real__lt a z /\ h4/realax/real__lt z b /\ h4/lim/diffl f l z /\ h4/real/real__sub (f b) (f a) = h4/realax/real__mul (h4/real/real__sub b a) l)
% 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
% Assm: h4/poly/POLY__DIFFERENTIABLE: !x l. h4/lim/differentiable (\x0. h4/poly/poly l x0) x
% Assm: h4/poly/POLY__CONT: !x l. h4/lim/contl (\x0. h4/poly/poly l x0) x
% Goal: !p b a. h4/realax/real__lt a b ==> (?x. h4/realax/real__lt a x /\ h4/realax/real__lt x b /\ h4/real/real__sub (h4/poly/poly p b) (h4/poly/poly p a) = h4/realax/real__mul (h4/real/real__sub b a) (h4/poly/poly (h4/poly/diff p) 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_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_lims_DIFFu_u_UNIQ]: !x m l f. h4/lim/diffl f l x /\ h4/lim/diffl f m x ==> l = m
% Assm [h4s_lims_MVT]: !f b a. h4/realax/real__lt a b /\ (!x. h4/real/real__lte a x /\ h4/real/real__lte x b ==> h4/lim/contl f x) /\ (!x. h4/realax/real__lt a x /\ h4/realax/real__lt x b ==> h4/lim/differentiable f x) ==> (?l z. h4/realax/real__lt a z /\ h4/realax/real__lt z b /\ h4/lim/diffl f l z /\ h4/real/real__sub (happ f b) (happ f a) = h4/realax/real__mul (h4/real/real__sub b a) l)
% 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)
% Assm [h4s_polys_POLYu_u_DIFFERENTIABLE]: !_0. (!l x0. happ (happ _0 l) x0 = h4/poly/poly l x0) ==> (!x l. h4/lim/differentiable (happ _0 l) x)
% Assm [h4s_polys_POLYu_u_CONT]: !_0. (!l x0. happ (happ _0 l) x0 = h4/poly/poly l x0) ==> (!x l. h4/lim/contl (happ _0 l) x)
% Goal: !p b a. h4/realax/real__lt a b ==> (?x. h4/realax/real__lt a x /\ h4/realax/real__lt x b /\ h4/real/real__sub (h4/poly/poly p b) (h4/poly/poly p a) = h4/realax/real__mul (h4/real/real__sub b a) (h4/poly/poly (h4/poly/diff p) 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_Q214319,TV_Q214315]: ![V_f, V_g]: (![V_x]: s(TV_Q214315,happ(s(t_fun(TV_Q214319,TV_Q214315),V_f),s(TV_Q214319,V_x))) = s(TV_Q214315,happ(s(t_fun(TV_Q214319,TV_Q214315),V_g),s(TV_Q214319,V_x))) => s(t_fun(TV_Q214319,TV_Q214315),V_f) = s(t_fun(TV_Q214319,TV_Q214315),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> 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_DIFFu_u_UNIQ, axiom, ![V_x, V_m, V_l, V_f]: ((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)))) & 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_m),s(t_h4s_realaxs_real,V_x))))) => s(t_h4s_realaxs_real,V_l) = s(t_h4s_realaxs_real,V_m))).
fof(ah4s_lims_MVT, axiom, ![V_f, V_b, V_a]: ((p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b)))) & (![V_x]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_b))))) => p(s(t_bool,h4s_lims_contl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x))))) & ![V_x]: ((p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_b))))) => 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, V_z]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_z)))) & (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_z),s(t_h4s_realaxs_real,V_b)))) & (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_z)))) & s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_b))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_a))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_b),s(t_h4s_realaxs_real,V_a))),s(t_h4s_realaxs_real,V_l)))))))).
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(ah4s_polys_POLYu_u_DIFFERENTIABLE, 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_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)))))).
fof(ah4s_polys_POLYu_u_CONT, 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_contl(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)))))).
fof(ch4s_polys_POLYu_u_MVT, conjecture, ![V_p, V_b, V_a]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_b)))) => ?[V_x]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_a),s(t_h4s_realaxs_real,V_x)))) & (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_b)))) & s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p),s(t_h4s_realaxs_real,V_b))),s(t_h4s_realaxs_real,h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p),s(t_h4s_realaxs_real,V_a))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_b),s(t_h4s_realaxs_real,V_a))),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_p))),s(t_h4s_realaxs_real,V_x))))))))).
