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
  the Tran of TS is Function implies ([x, [y1, z]] in ==>.-relation(TS)
  & [x, [y2, z]] in ==>.-relation(TS) implies y1 = y2)
proof
  assume
A1: the Tran of TS is Function;
  assume that
A2: [x, [y1, z]] in ==>.-relation(TS) and
A3: [x, [y2, z]] in ==>.-relation(TS);
  consider s, v, t, w such that
A4: x = [s, v] and
  [y1, z] = [t, w] by A2,Th31;
  s, v ==>. y1, z, TS & s, v ==>. y2, z, TS by A2,A3,A4,Def4;
  hence thesis by A1,Th27;
end;
