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 Th34:
  for G1 being addVertices of G2, V
  holds G1 is c-vcolorable iff G2 is c-vcolorable
proof
  let G1 be addVertices of G2, V;
  hereby
    assume A1: G1 is c-vcolorable;
    G2 is Subgraph of G1 by GLIB_006:57;
    hence G2 is c-vcolorable by A1, Th31;
  end;
  assume G2 is c-vcolorable;
  :: we choose a proper coloring and color the added vertices
  :: in any color of those already taken
  then consider f2 being VColoring of G2 such that
    A2: f2 is proper & card rng f2 c= c;
  set x = the Element of rng f2;
  set h = (V \ the_Vertices_of G2) --> x, f1 = f2 +* h;
  A3: dom h = V \ the_Vertices_of G2;
  then reconsider f1 as VColoring of G1 by Th4;
  A4: f1 is proper by A2, A3, Th19;
  card rng f1 c= card rng f2 by Lm3, CARD_1:11;
  hence thesis by A2, A4, XBOOLE_1:1;
end;
