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
  TS is deterministic implies ([x, [y1, z1]] in ==>.-relation(TS) & [x,
  [y2, z2]] in ==>.-relation(TS) implies y1 = y2 & z1 = z2)
proof
  assume
A1: TS is deterministic;
  assume
  [x, [y1, z1]] in ==>.-relation(TS) & [x, [y2, z2]] in ==>.-relation( TS);
  then [y1, z1] = [y2, z2] by A1,Th44;
  hence thesis by XTUPLE_0:1;
end;
