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 Th27:
  for TS being transition-system over F st the Tran of TS is
  Function holds x1, x2 ==>. y1, z, TS & x1, x2 ==>. y2, z, TS implies y1 = y2
proof
  let TS be transition-system over F such that
A1: the Tran of TS is Function;
  assume that
A2: x1, x2 ==>. y1, z, TS and
A3: x1, x2 ==>. y2, z, TS;
  consider v1, w1 such that
A4: v1 = z and
A5: x1, w1 -->. y1, TS and
A6: x2 = w1^v1 by A2;
  consider v2, w2 such that
A7: v2 = z and
A8: x1, w2 -->. y2, TS and
A9: x2 = w2^v2 by A3;
  w1 = w2 by A4,A6,A7,A9,AFINSQ_1:28;
  hence thesis by A1,A5,A8,Th17;
end;
