reserve X,Y,z,s for set, L,L1,L2,A,B for List of X, x for Element of X,
  O,O1,O2,O3 for Operation of X, a,b,y for Element of X, n,m for Nat;
reserve F,F1,F2 for filtering Operation of X;

theorem
  L|F = L WHERE F
  proof
    thus L|F c= L WHERE F
    proof
      let z be object;
      assume z in L|F; then
      consider y such that
A1:   z in y.F & y in L by Th23;
      z = y by A1,Th46;
      hence thesis by A1,Th3;
    end;
    let z be object;
    assume
A2: z in L WHERE F; then
    reconsider x = z as Element of X;
A3: x in L & x.F <> {} by A2,Th3;
    set y = the Element of x.F;
A4: [x,y] in F by A3,RELAT_1:169;
    F c= id X by Def21; then
    x = y by A4,RELAT_1:def 10;
    hence thesis by A3,Th23;
  end;
