1.Introduction.- 2.Higher order Derivatives of Dirichlets' Energy.- 3.Very Special Case; The Theorem for n + 1 Even and m + 1 Odd .- 4.The First Main Theorem; Non-Exceptional Branch Points.- 5.The Second Main Theorem: Exceptional Branch Points; The Condition k > l.- 6.Exceptional Branch Points Without The Condition k > l.- 7.New Brief Proofs of the Gulliver-Osserman-Royden Theorem .- 8.Boundary Branch Points.- Scholia.- Appendix.- Bibliography....