The reference is given in the paper you cite:
M. Eudave-Mun ̃oz, Primeness and sums of tangles, Trans. Am. Math. Soc. 306, 773-790 (1988)

The arguments are purely combinatorial, but there should be a simplification using sutured manifold theory along the lines of Scharlemann and Thompson's paper "Unknotting number, genus, and companion tori" by Scharlemann and Thompson MR0929535

Off the top of my head, I don't see a way to convert either type of argument into a something about diagrams.