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 Th12:
  for f9 being one-to-one Function, f2 being VColoring of G
  st f2 = f9*f & f is proper & rng f c= dom f9 holds f2 is proper
proof
  let f9 be one-to-one Function, f2 be VColoring of G;
  assume A1: f2 = f9*f & f is proper & rng f c= dom f9;
  now
    let e,v,w be object;
    assume A2: e Joins v,w,G;
    then v in the_Vertices_of G & w in the_Vertices_of G by GLIB_000:13;
    then A3: v in dom f & w in dom f by PARTFUN1:def 2;
    then A4: f2.v = f9.(f.v) & f2.w = f9.(f.w) by A1, FUNCT_1:13;
    A5: f.v in rng f & f.w in rng f by A3, FUNCT_1:3;
    f.v <> f.w by A1, A2, Th10;
    hence f2.v <> f2.w by A1, A4, A5, FUNCT_1:def 4;
  end;
  hence thesis by Th10;
end;
