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 Th19:
  for TS being deterministic transition-system over F holds u <> v
  & x, u -->. z1, TS & x, v -->. z2, TS implies not ex w st u^w = v or v^w = u
proof
  let TS be deterministic transition-system over F;
  assume that
A1: u <> v and
A2: x, u -->. z1, TS and
A3: x, v -->. z2, TS;
  x in TS by A2,Th15;
  then reconsider x as Element of TS;
  [[x, v], z2] in the Tran of TS by A3;
  then
A4: [x, v] in dom the Tran of TS by XTUPLE_0:def 12;
  [[x, u], z1] in the Tran of TS by A2;
  then [x, u] in dom the Tran of TS by XTUPLE_0:def 12;
  hence thesis by A1,A4,Def1;
end;
