reserve X1,X2,X3,X4 for set;

theorem LemY:
  for X be set,
      S be with_empty_element Subset-Family of X holds
    DIFFERENCE (S,S) = the set of all A \ B where A, B is Element of S
  proof
    let X be set,
        S be with_empty_element Subset-Family of X;
    thus DIFFERENCE (S,S) c= the set of all A \ B where A, B is Element of S
    proof
      let x be object;
      assume x in DIFFERENCE (S,S); then
      consider X,Y being set such that
A1:   X in S & Y in S & x = X \ Y by SETFAM_1:def 6;
      thus thesis by A1;
    end;
    let x be object;
    assume x in the set of all A \ B where A, B is Element of S; then
    consider A1, B1 being Element of S such that
A2: x = A1 \ B1;
    thus thesis by A2,SETFAM_1:def 6;
  end;
