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;
reserve t for TColoring of G;

theorem Th182:
  for v,e,w being object, G1 being addAdjVertex of G2,v,e,w
  holds G1 is finite-tcolorable iff G2 is finite-tcolorable
proof
  let v,e,w be object, G1 be addAdjVertex of G2,v,e,w;
  hereby
    assume A1: G1 is finite-tcolorable;
    G2 is Subgraph of G1 by GLIB_006:57;
    hence G2 is finite-tcolorable by A1;
  end;
  assume G2 is finite-tcolorable;
  then consider n such that
    A2: G2 is n-tcolorable;
  per cases;
  suppose G2 is non edgeless;
    then G1 is (n+`1)-tcolorable by A2, Th172;
    hence thesis;
  end;
  suppose G2 is edgeless;
    then G1 is 3-tcolorable by Th174;
    hence thesis;
  end;
end;
