Postulation of general unions of lines and +lines in positive characteristic

A +line is a scheme R ⊂ ℙr with a line as its reduction L = Rred which is the union of L and a tangent vector v ⊈ L with vred ∈ L. Here we prove in arbitrary characteristic that for r ≥ 4 a general union of lines and +lines has maximal rank. We use the case r = 3 proved by myself in a previous paper and adapt the characteristic zero proof of the case r > 3 given in the same paper.