Various proofs of Sylvester's (determinant) identity

Despite the fact that the importance of Sylvester's determinant identity has been recognized in the past, we were able to find only one proof of it in English (Bareiss, 1968), with reference to some others. (Recall that Sylvester (1857) stated this theorem without proof.) Having used this identity, recently, in the validity proof of our new, improved, matrix-triangularization subresultant polynomial remainder sequence method (Akritas et al., 1995), we decided to collect all the proofs we found of this identity-one in English, four in German and two in Russian, in that order-in a single paper (Akritas et al., 1992). It turns out that the proof in English is identical to an earlier one in German. Due to space limitations two proofs are omitted.