proposition: Lemma derivable_pt_lim_comp_2d : forall f1 f2 f3 x l1x l1y l2 l3, differentiable_pt_lim f1 (f2 x) (f3 x) l1x l1y -> derivable_pt_lim f2 x l2 -> derivable_pt_lim f3 x l3 -> derivable_pt_lim (fun t => f1 (f2 t) (f3 t)) x (l1x * l2 + l1y * l3).
proof: Proof. intros. apply (differentiable_pt_lim_proj1_1 _ x 0 (l1x * l2 + l1y * l3)). pattern 0 at 2 ; replace 0 with (l1x * 0 + l1y * 0) by ring. apply differentiable_pt_lim_comp. apply H. apply: differentiable_pt_lim_proj1_0 H0. apply: differentiable_pt_lim_proj1_0 H1. Qed.
imports: Coq(Reals Psatz ssreflect) Rcomplements Hierarchy Continuity Derive