Lemma is_derive_RInt_param_bound_comp_aux1 : forall (f : R -> R -> R) (a : R -> R) (x : R), (exists eps:posreal, locally x (fun y : R => ex_RInt (fun t => f y t) (a x - eps) (a x + eps))) -> (exists eps:posreal, locally x (fun x0 : R => forall t : R, a x-eps <= t <= a x+eps -> ex_derive (fun u : R => f u t) x0)) -> (locally_2d (fun x' t => continuity_2d_pt (fun u v : R => Derive (fun z : R => f z v) u) x' t) x (a x)) -> continuity_2d_pt (fun u v : R => Derive (fun z : R => RInt (fun t : R => f z t) v (a x)) u) x (a x).