
theorem
  for G1, G2 being _Graph, W1 being Walk of G1, W2 being Walk of G2
  st W1 = W2 holds W1 is vertex-distinct implies W2 is vertex-distinct
proof
  let G1, G2 be _Graph;
  let W1 be Walk of G1, W2 be Walk of G2;
  assume A1: W1 = W2;
  assume W1 is vertex-distinct;
  then for m,n being odd Element of NAT st
    m <= len W2 & n <= len W2 & W2.m = W2.n holds m = n by A1, GLIB_001:def 29;
  hence W2 is vertex-distinct by GLIB_001:def 29;
end;
