reserve X for set;
reserve UN for Universe;

theorem
  for u being Element of UN holds
  disjoint-union(u,u) = { [ u , {{}} ] }
  proof
    let u be Element of UN;
    disjoint-union(u,u) = u .--> {{}UN} by FUNCT_4:81
                       .= { [u,{{}UN}] } by FUNCT_4:82;
    hence thesis;
  end;
