
theorem
  for G being _Graph, W being Walk of G, e,x being object
  st e Joins W.last(),x,G holds W.addEdge(e).length() = W.length() + 1
proof
  let G be _Graph, W be Walk of G, e,x be object;
  assume A1: e Joins W.last(),x,G;
  2*W.addEdge(e).length()+1 = len W.addEdge(e) by GLIB_001:112
    .= len W + 2 by A1, GLIB_001:64
    .= 2*W.length()+1 + 2 by GLIB_001:112
    .= 2*(W.length()+1) + 1;
  hence thesis;
end;
