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 Th49:
  for v being object, G1 being addAdjVertexAll of G2,v,V
  holds G1 is finite-vcolorable iff G2 is finite-vcolorable
proof
  let v be object, G1 be addAdjVertexAll of G2,v,V;
  thus G1 is finite-vcolorable implies G2 is finite-vcolorable;
  assume G2 is finite-vcolorable;
  then consider n such that
    A1: G2 is n-vcolorable;
  G1 is (n+`1)-vcolorable by A1, Th39;
  hence thesis;
end;
