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 Th40:
  for G2 being removeParallelEdges of G1
  holds G1 is c-vcolorable iff G2 is c-vcolorable
proof
  let G2 be removeParallelEdges of G1;
  thus G1 is c-vcolorable implies G2 is c-vcolorable by Th31;
  assume G2 is c-vcolorable;
  then consider f2 being VColoring of G2 such that
    A1: f2 is proper & card rng f2 c= c;
  the_Vertices_of G1 = the_Vertices_of G2 by GLIB_000:def 33;
  then reconsider f1 = f2 as VColoring of G1;
  now
    let v,w be Vertex of G1;
    reconsider v9=v, w9=w as Vertex of G2 by GLIB_000:def 33;
    assume v,w are_adjacent;
    then v9,w9 are_adjacent by GLIBPRE1:75;
    hence f1.v <> f1.w by A1;
  end;
  then f1 is proper;
  hence thesis by A1;
end;
