
theorem
  for G being WVGraph, v,x being set holds the_Weight_of G =
  the_Weight_of G.labelVertex(v,x)
proof
  let G be WVGraph, e,x be set;
  set G2 = G.labelVertex(e,x);
  now
    per cases;
    suppose
      e in the_Vertices_of G;
      then G2 = G.set(VLabelSelector, the_VLabel_of G +* (e.-->x)) by Def22;
      hence thesis by GLIB_000:9;
    end;
    suppose
      not e in the_Vertices_of G;
      hence thesis by Def22;
    end;
  end;
  hence thesis;
end;
