theorem
  A <> {} implies 0**A = {0}
proof
  assume
A1: A <> {};
A2: {0} c= 0**A
  proof
    set x = the Element of A;
    let a be Nat;
    assume
A3: a in {0};
A4: 0*x in {0*c: c in A} by A1;
    0**A = {0*c: c in A} by Th194;
    hence thesis by A3,A4,TARSKI:def 1;
  end;
  0**A c= {0} by Th201;
  hence thesis by A2;
end;
