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
  for v,e,w being object, G1 being addEdge of G2,v,e,w
  st v <> w holds G1 is finite-tcolorable iff G2 is finite-tcolorable
proof
  let v,e,w be object, G1 be addEdge of G2,v,e,w;
  assume A1: v <> w;
  hereby
    assume A2: G1 is finite-tcolorable;
    G2 is Subgraph of G1 by GLIB_006:57;
    hence G2 is finite-tcolorable by A2;
  end;
  assume G2 is finite-tcolorable;
  then consider n such that
    A3: G2 is n-tcolorable;
  G1 is (n+`2)-tcolorable by A1, A3, Th171;
  hence thesis;
end;
