The map *f* is essentially unique: if *z*_{0} is an element of *U* and φ in (-π, π] is an arbitrary angle, then there exists precisely one *f* as above with the additional properties *f*(*z*_{0}) = 0 and arg *f* '(*z*_{0}) = φ.

As a corollary, any two such simply connected open sets (which are different from **C** and **C** U {∞}) can be conformally mapped into each other.

The theorem was proved by Bernhard Riemann in 1851, but his proof depended on a statement in the calculus of variations which was only later proven by David Hilbert.