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
  for v,w being Vertex of G2, e being object, G1 being addEdge of G2,v,e,w
  st v,w are_adjacent holds G1 is c-vcolorable iff G2 is c-vcolorable
proof
  let v,w be Vertex of G2, e be object, G1 be addEdge of G2,v,e,w;
  assume A1: v,w are_adjacent;
  hereby
    assume A2: G1 is c-vcolorable;
    G2 is Subgraph of G1 by GLIB_006:57;
    hence G2 is c-vcolorable by A2, Th31;
  end;
  assume G2 is c-vcolorable;
  then consider f2 being VColoring of G2 such that
    A3: f2 is proper & card rng f2 c= c;
  the_Vertices_of G1 = the_Vertices_of G2 by GLIB_006:102;
  then reconsider f1 = f2 as VColoring of G1;
  f1 is proper by A1, A3, Th20;
  hence thesis by A3;
end;
