
theorem Th13:
  for G being _Graph for W1, W2 being Walk of G st W1
  is_a_prefix_of W2 holds W1.vertices() c= W2.vertices()
proof
  let G be _Graph, W1, W2 be Walk of G such that
A1: W1 c= W2;
  let x be object;
  assume x in W1.vertices();
  then consider n being odd Element of NAT such that
A2: n <= len W1 and
A3: W1.n = x by GLIB_001:87;
  1 <= n by ABIAN:12;
  then n in dom W1 by A2,FINSEQ_3:25;
  then
A4: W2.n = x by A1,A3,GRFUNC_1:2;
  len W1 <= len W2 by A1,FINSEQ_1:63;
  then n <= len W2 by A2,XXREAL_0:2;
  hence thesis by A4,GLIB_001:87;
end;
