reserve G,G1,G2 for _Graph;
reserve W,W1,W2 for Walk of G;
reserve e,x,y,z for set;
reserve v for Vertex of G;
reserve n,m for Element of NAT;

theorem
  for W1 being Walk of G1, W2 being Walk of G2 st G1 == G2 & W1 = W2
holds (W1 is closed iff W2 is closed) & (W1 is directed iff W2 is directed) & (
W1 is trivial iff W2 is trivial) & (W1 is Trail-like iff W2 is Trail-like) & (
  W1 is Path-like iff W2 is Path-like) & (W1 is vertex-distinct iff W2 is
  vertex-distinct)
proof
  let W1 be Walk of G1, W2 be Walk of G2;
  assume that
A1: G1 == G2 and
A2: W1 = W2;
  G1 is Subgraph of G2 by A1,GLIB_000:87;
  hence thesis by A2,Th174;
end;
