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 Th11:
  f is proper iff for e,v,w being object st e DJoins v,w,G holds f.v <> f.w
proof
  hereby
    assume A1: f is proper;
    let e,v,w be object;
    assume e DJoins v,w,G;
    then e Joins v,w,G by GLIB_000:16;
    hence f.v <> f.w by A1, Th10;
  end;
  assume A2: for e,v,w being object st e DJoins v,w,G holds f.v <> f.w;
  now
    let e,v,w be object;
    assume e Joins v,w,G;
    then per cases by GLIB_000:16;
    suppose e DJoins v,w,G;
      hence f.v <> f.w by A2;
    end;
    suppose e DJoins w,v,G;
      hence f.v <> f.w by A2;
    end;
  end;
  hence thesis by Th10;
end;
