reserve x, x1, x2, y, y1, y2, z, z1, z2 for object, X, X1, X2 for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u9, u1, u2, v, v1, v2, w, w1, w2 for Element of E^omega;
reserve F, F1, F2 for Subset of E^omega;
reserve i, k, l, n for Nat;

theorem
  for TS being transition-system over F holds dom (the Tran of TS)
  = {} implies TS is deterministic
proof
  let TS be transition-system over F;
  assume dom (the Tran of TS) = {};
  then
  (the Tran of TS) = {} & for s being Element of TS, u, v st u <> v & [s,
u] in dom (the Tran of TS) & [s, v] in dom (the Tran of TS) holds not ex w st u
  ^w = v or v^w = u;
  hence thesis;
end;
