Embedded o-minimal structures

We prove the following two theorems on embedded o-minimal structures: Theorem 1.Let ℳ ≺N be o-minimal structures and let ℳ* be the expansion of ℳ by all traces in M of 1-variable formulas in , that is all sets of the form φ(M, ā) for ā ⊆ N and φ(x, ȳ) ∈ ℒ(N). Then, for any N-formula Ψ(x₁, ⋯, xk), the set Ψ(M^k) is ℳ*-definable.Theorem 2.Let be an 1-saturated structure and let S be a sort in N^eq. Let be the N-induced structure on S and assume that is o-minimal. Then is stably embedded.