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 Th30:
  for TS being deterministic transition-system over F holds x1, x2
  ==>. y1, z1, TS & x1, x2 ==>. y2, z2, TS implies y1 = y2 & z1 = z2
proof
  let TS be deterministic transition-system over F;
  assume that
A1: x1, x2 ==>. y1, z1, TS and
A2: x1, x2 ==>. y2, z2, TS;
  consider v2, w2 such that
A3: v2 = z2 and
A4: x1, w2 -->. y2, TS and
A5: x2 = w2^v2 by A2;
  consider v1, w1 such that
A6: v1 = z1 and
A7: x1, w1 -->. y1, TS and
A8: x2 = w1^v1 by A1;
A9: the Tran of TS is Function by Def1;
  (ex u9 st w1^u9 = w2 & v1 = u9^v2) or ex u9 st w2^u9 = w1 & v2 = u9^v1
  by A8,A5,Th13;
  then w1 = w2 by A7,A4,Th19;
  then v1 = v2 by A8,A5,AFINSQ_1:28;
  hence thesis by A1,A2,A6,A3,A9,Th27;
end;
