I must admit I have never read this reference, but I remember it from a similar discussion on a German forum, according to which there is a simple proof in
Michiel Hazewinkel, A short elementary proof of Grothendieck's theorem on algebraic vectorbundles over the projective line, Journal of pure and applied algebra 25 (1982), pp. 207 - 211Michiel Hazewinkel and Clyde Martin, A short elementary proof of Grothendieck's theorem on algebraic vectorbundles over the projective line, Journal of pure and applied algebra 25 (1982), pp. 207 - 211.