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 Th19:
  for G1 being addVertices of G2, V
  for f1 being VColoring of G1, f2 being VColoring of G2, h being Function
  st dom h = V \ the_Vertices_of G2 & f1 = f2 +* h & f2 is proper
  holds f1 is proper
proof
  let G1 be addVertices of G2, V;
  let f1 be VColoring of G1, f2 be VColoring of G2, h being Function;
  assume A1: dom h = V \ the_Vertices_of G2 & f1 = f2 +* h & f2 is proper;
  now
    let e,v,w be object;
    assume e Joins v,w,G1;
    then A2: e Joins v,w,G2 by GLIB_006:87;
    v in the_Vertices_of G2 & w in the_Vertices_of G2 by A2, GLIB_000:13;
    then not v in dom h & not w in dom h by A1, XBOOLE_0:def 5;
    then f1.v = f2.v & f1.w = f2.w by A1, FUNCT_4:11;
    hence f1.v <> f1.w by A1, A2, Th10;
  end;
  hence f1 is proper by Th10;
end;
