  The notions of N=1 Neveu-Schwarz vertex operator superalgebra over a
Grassmann algebra and with odd formal variables and of N=1 Neveu-Schwarz vertex
operator superalgebra over a Grassmann algebra and without odd formal variables
are introduced, and we show that the respective categories of such objects are
isomorphic. The weak supercommutativity and weak associativity properties for
an N=1 Neveu-Schwarz vertex operator superalgebra with odd formal variables are
established, and we show that in the presence of the other axioms, weak
supercommutativity and weak associativity are equivalent to the Jacobi
identity. In addition, we prove the supercommutativity and associativity
properties for an N=1 Neveu-Schwarz vertex operator superalgebra with odd
formal variables and show that in the presence of the other axioms,
supercommutativity and associativity are equivalent to the Jacobi identity.
