
theorem
  for G being _Graph, W being Walk of G, v being Vertex of G
  st v in W.vertices() holds G.walkOf(v) is_substring_of W, 0
proof
  let G be _Graph, W be Walk of G, v be Vertex of G;
  assume v in W.vertices();
  then consider n being odd Element of NAT such that
    A1: n <= len W & W.n = v by GLIB_001:87;
  now
    assume len G.walkOf(v) > 0;
    take n;
    thus 0 <= n;
    thus n <= len W by A1;
    A3: (n-'1)+len G.walkOf(v) = n-'1+1 by GLIB_001:13
      .= n+1-'1 by ABIAN:12, NAT_D:38
      .= n+1-1 by NAT_D:37
      .= n;
    1 <= n by ABIAN:12;
    then A4: n in dom W by A1, FINSEQ_3:25;
    thus mid(W,n,(n-'1)+len G.walkOf(v)) = <*W.n*> by A3, A4, Th7
      .= G.walkOf(v) by A1, GLIB_001:def 4;
  end;
  hence thesis by FINSEQ_8:def 7;
end;
