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 Th29:
  for TS being transition-system over F st not <%>E in rng dom (
  the Tran of TS) holds x, u ==>. y, v, TS implies len u > len v
proof
  let TS be transition-system over F such that
A1: not <%>E in rng dom (the Tran of TS);
  assume
A2: x, u ==>. y, v, TS;
  then consider w such that
A3: x, w -->. y, TS and
A4: u = w^v;
A5: w in rng dom (the Tran of TS) by A3,Th15;
  per cases;
  suppose
A6: v = <%>E;
    then u <> <%>E by A1,A2,Th28;
    then len u > 0;
    hence thesis by A6;
  end;
  suppose
    v <> <%>E;
    hence thesis by A1,A4,A5,Th10;
  end;
end;
