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

theorem Th116:
  for v,e,w being object, G1 being addAdjVertex of G2,v,e,w
  holds G1 is finite-ecolorable iff G2 is finite-ecolorable
proof
  let v,e,w be object, G1 be addAdjVertex of G2,v,e,w;
  hereby
    assume A1: G1 is finite-ecolorable;
    G2 is Subgraph of G1 by GLIB_006:57;
    hence G2 is finite-ecolorable by A1;
  end;
  assume G2 is finite-ecolorable;
  then consider n such that
    A2: G2 is n-ecolorable;
  reconsider G2 as n-ecolorable _Graph by A2;
  G1 is addAdjVertex of G2,v,e,w;
  hence thesis;
end;
