reserve E,V for set, G,G1,G2 for _Graph, c,c1,c2 for Cardinal, n for Nat;
reserve f for VColoring of G;

theorem Th16:
  for f1 being VColoring of G1, f2 being VColoring of G2
  st G1 == G2 & f1 = f2 & f1 is proper holds f2 is proper
proof
  let f1 be VColoring of G1, f2 be VColoring of G2;
  assume A1: G1 == G2 & f1 = f2 & f1 is proper;
  now
    let e,v,w be object;
    assume e Joins v,w,G2;
    then e Joins v,w,G1 by A1, GLIB_000:88;
    hence f2.v <> f2.w by A1, Th10;
  end;
  hence thesis by Th10;
end;
