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 Th4:
  for G1 being addVertices of G2, V, f being VColoring of G2
  for h being Function st dom h = V \ the_Vertices_of G2
  holds f +* h is VColoring of G1
proof
  let G1 be addVertices of G2, V;
  let f be VColoring of G2, h being Function;
  assume A1: dom h = V \ the_Vertices_of G2;
  dom(f+*h) = dom f \/ dom h by FUNCT_4:def 1
    .= the_Vertices_of G2 \/ (V \ the_Vertices_of G2) by A1, PARTFUN1:def 2
    .= the_Vertices_of G2 \/ V by XBOOLE_1:39
    .= the_Vertices_of G1 by GLIB_006:def 10;
  hence f+*h is VColoring of G1 by RELAT_1:def 18, PARTFUN1:def 2;
end;
