theorem Th2:
  Cl(A) <> {} iff A <> {}
proof
  A <> {} implies Cl(A) <> {}
  proof
    set x = the Element of A;
A1: A c= Cl A by PRE_TOPC:18;
    assume
A2: A <> {};
    ex x be set st x in Cl A
    proof
      take x;
      thus thesis by A2,A1;
    end;
    hence thesis;
  end;
  hence thesis by PRE_TOPC:22;
end;
