His proof was adapted to the smooth case by Morse and Emilio Baiada. The essential property was used by Cerf in order to prove that every orientation-preserving diffeomorphism of  is isotopic to the identity, seen as a one-parameter extension of the Schoenflies theorem for .
Morse and Emilio Baiada refined his proof for the smooth case. They utilized a key characteristic to demonstrate that every orientation-preserving diffeomorphism of  is isotopic to the identity, as a one-parameter extension of the Schoenflies theorem.