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 Th117:
  for v being object, V being finite set, G1 being addAdjVertexAll of G2,v,V
  holds G1 is finite-ecolorable iff G2 is finite-ecolorable
proof
  let v be object, V be finite set, G1 be addAdjVertexAll of G2,v,V;
  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 addAdjVertexAll of G2,v,V;
  hence thesis;
end;
