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;
reserve TS for non empty transition-system over F;
reserve s, s9, s1, s2, t, t1, t2 for Element of TS;
reserve S for Subset of TS;

theorem Th64:
  not <%>E in rng dom (the Tran of TS) implies for P being
RedSequence of ==>.-relation(TS) st P.1 = [x, v] & P.len P = [y, w] holds len v
  > len w or len P = 1 & x = y & v = w
proof
  assume
A1: not <%>E in rng dom (the Tran of TS);
  let P be RedSequence of ==>.-relation(TS) such that
A2: P.1 = [x, v] & P.len P = [y, w];
  consider u such that
A3: v = u^w by A2,Th53;
A4: len v >= len w by A2,Th59;
  per cases;
  suppose
    len v > len w;
    hence thesis;
  end;
  suppose
A5: len v <= len w;
A6: len v = len u + len w by A3,AFINSQ_1:17;
    len v = len w by A4,A5,XXREAL_0:1;
    then u = <%>E by A6;
    hence thesis by A1,A2,Th60,A3;
  end;
end;
