reserve x, y, X for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u1, v, v1, v2, w, w9, w1, w2 for Element of E^omega;
reserve F for Subset of E^omega;
reserve i, k, l for Nat;
reserve TS for non empty transition-system over F;
reserve S, T for Subset of TS;
reserve SA for non empty semiautomaton over F;
reserve A for non empty automaton over F;
reserve p, q for Element of A;

theorem Th18:
  w in Lang(A) iff ex p, q st p in the InitS of A & q in the
  FinalS of A & p, w ==>* q, A
proof
  thus w in Lang(A) implies ex p, q st p in the InitS of A & q in the FinalS
  of A & p, w ==>* q, A
  proof
    assume w in Lang(A);
    then
    ex u st w = u & ex p, q st p in the InitS of A & q in the FinalS of A
    & p, u ==>* q, A;
    hence thesis;
  end;
  thus thesis;
end;
