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
  for v,e,w being object, G1 being addEdge of G2,v,e,w st v <> w
  holds G1 is finite-vcolorable iff G2 is finite-vcolorable
proof
  let v,e,w be object, G1 be addEdge of G2,v,e,w;
  assume A1: v <> w;
  thus G1 is finite-vcolorable implies G2 is finite-vcolorable;
  assume G2 is finite-vcolorable;
  then consider n such that
    A2: G2 is n-vcolorable;
  G1 is (n+`1)-vcolorable by A1, A2, Th36;
  hence thesis;
end;
