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 Th48:
  for v,e,w being object, G1 being addAdjVertex of G2,v,e,w
  holds G1 is finite-vcolorable iff G2 is finite-vcolorable
proof
  let v,e,w be object, G1 be addAdjVertex of G2,v,e,w;
  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;
  per cases;
  suppose G2 is non edgeless;
    then G1 is n-vcolorable by A1, Th37;
    hence thesis;
  end;
  suppose G2 is edgeless;
    then G1 is 2-vcolorable by Th38;
    hence thesis;
  end;
end;
