theorem
  for X being Subset of T st X = {} holds 0(.)X = {}
proof
  let X be Subset of T;
  assume
A1: X = {};
  now
    given x being object such that
A2: x in 0(.)X;
    ex a being Point of T st x = 0 * a & a in X by A2;
    hence contradiction by A1;
  end;
  hence thesis by XBOOLE_0:def 1;
end;
