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;
reserve i for Element of NAT;

theorem
  for A being FinSequence of bool X holds
  for n1,n2,m1,m2 being Nat st n1 <= m1 & n2 <= m2 holds
  ROUGH(A,m1,n2) c= ROUGH(A,n1,m2)
  proof
    let A be FinSequence of bool X;
    let n1,n2,m1,m2 be Nat;
    assume A1: n1 <= m1;
    assume A2: n2 <= m2;
    let z be object; assume
A3: z in ROUGH(A,m1,n2); then
    z in {x: m1 <= #occurrences(x,A) & #occurrences(x,A) <= n2} by Def25; then
    consider a such that
A4: z = a & m1 <= #occurrences(a,A) & #occurrences(a,A) <= n2;
    n1 <= #occurrences(a,A) & #occurrences(a,A) <= m2
    by A1,A2,A4,XXREAL_0:2; then
    z in {x: n1 <= #occurrences(x,A) & #occurrences(x,A) <= m2} by A4;
    hence z in ROUGH(A,n1,m2) by A3,Def25;
  end;
