  In 1853 J. Sylvester introduced a family of double sum expressions for two
finite sets of indeterminates and showed that some members of the family are
essentially the polynomial subresultants of the monic polynomials associated
with these sets. In 2009, in a joint work with C. D'Andrea and H. Hong we gave
the complete description of all the members of the family as expressions in the
coefficients of these polynomials. In 2010, M.-F. Roy and A. Szpirglas
presented a new and natural inductive proof for the cases considered by
Sylvester. Here we show how induction also allows to obtain the full
description of Sylvester's double-sums.
