theorem
  for D being Subset-Family of X, A being Subset of D holds
    union A is Subset of X
proof
  let D be Subset-Family of X, A be Subset of D;
  union A c= X
  proof
    let e be object;
    assume e in union A;
    then ex u being set st e in u & u in A by TARSKI:def 4;
    then e in union D by TARSKI:def 4;
    hence thesis;
  end;
  hence thesis;
end;
