Equivalence of Deterministic and Nondeterministic Epsilon Automata

For simplicity, we adopt the following convention: x, y, X denote sets, E denotes a non empty set, e denotes an element of E, u, u1, v, v1, v2, w denote elements of Eω, F denotes a subset of Eω, i, k, l denote natural numbers, T denotes a non empty transition-system over F , and S, T denote subsets of T. One can prove the following propositions: (1) If i ≥ k + l, then i ≥ k. (2) For all finite sequences a, b such that a a b = a or b a a = a holds b = ∅. (3) For all finite sequences p, q such that k ∈ dom p and len p + 1 = len q holds k + 1 ∈ dom q. (4) If lenu = 1, then there exists e such that 〈e〉 = u and e = u(0).