It depends on what do you mean by "direct". There is no elementary proof, as far as I known, even for polynomials when $d=2$, see page 8 of this note. When $d=1$, the statement is an easy exercise: one can write $f_1= da, f_2=db$ with $(a,b) = R$ since $R$ is an UFD.

There are a few proofs of this very interesting theorem, analytic (the original one, over complex numbers), using duality theory (Lipman-Sathaye, for all regular rings) and reduction to characteristic $p$ + tight closure (Hochster-Huneke, when $R$ contains a field, see the note in the first paragraph for some exposition of this method) but I am not sure any of them can be called direct. Any new proof will be very exciting!

It depends on what do you mean by "direct". There is no elementary proof, as far as I known, even for polynomials when $d=2$, see page 8 of this note. When $d=1$, the statement is an easy exercise: one can write $f_1= da, f_2=db$ with $(a,b) = R$ since $R$ is an UFD.

There are a few proofs of this very interesting theorem, analytic (the original one, over complex numbers), using duality theory (Lipman-Sathaye, for all regular rings) and reduction to characteristic $p$ + tight closure (Hochster-Huneke, when $R$ contains a field, see the note in the first paragraph for some exposition of this method) but I am not sure any of them can be called direct. Any new proof will be very exciting!