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 Th23:
  for TS being transition-system over F holds x, y -->. z, TS iff
  x, y ==>. z, <%>E, TS
proof
  let TS be transition-system over F;
  thus x, y -->. z, TS implies x, y ==>. z, <%>E, TS
  proof
    assume
A1: x, y -->. z, TS;
    then y in F by Th15;
    then reconsider w = y as Element of E^omega;
    w = w^{};
    hence thesis by A1;
  end;
  assume x, y ==>. z, <%>E, TS;
  then ex v, w st v = <%>E & x, w -->. z, TS & y = w^v;
  hence thesis;
end;
