reserve X1,X2,X3,X4 for set;

theorem LemX1:
  for X be set,
      S be with_empty_element Subset-Family of X,
      x be object st x in semidiff S holds
    ex A, B being Element of S st x = A \ B
  proof
    let X be set,
        S be with_empty_element Subset-Family of X,
        x be object;
    assume x in semidiff S; then
    x in the set of all A \ B where A, B is Element of S by LemY;
    hence thesis;
  end;
