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
  for TS being transition-system over F st not <%>E in rng dom (the Tran
  of TS) holds not x, <%>E -->. y, TS
proof
  let TS be transition-system over F such that
A1: not <%>E in rng dom (the Tran of TS);
  assume x, <%>E -->. y, TS;
  then [[x, <%>E], y] in the Tran of TS;
  then [x, <%>E] in dom (the Tran of TS) by XTUPLE_0:def 12;
  hence contradiction by A1,XTUPLE_0:def 13;
end;
