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;
reserve TS for non empty transition-system over F;
reserve s, s9, s1, s2, t, t1, t2 for Element of TS;
reserve S for Subset of TS;

theorem Th91:
  not <%>E in rng dom (the Tran of TS) implies (x, u ==>* y, u, TS iff x = y)
proof
  assume
A1: not <%>E in rng dom (the Tran of TS);
  thus x, u ==>* y, u, TS implies x = y
  proof
    assume x, u ==>* y, u, TS;
    then ==>.-relation(TS) reduces [x, u], [y, u];
    then
    ex p being RedSequence of ==>.-relation(TS) st p.1 = [x, u] & p.len p =
    [y, u] by REWRITE1:def 3;
    hence thesis by A1,Th60;
  end;
  assume x = y;
  hence thesis by Th82;
end;
