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
  n <= m implies ROUGH(A,m) c= ROUGH(A,n)
  proof
    let A be FinSequence of bool X;
    assume A1: n <= m;
    let z be object; assume
A2: z in ROUGH(A,m); then
    z in {x: m <= #occurrences(x,A)} by Def24; then
    consider a such that
A3: z = a & m <= #occurrences(a,A);
    n <= #occurrences(a,A) by A1,A3,XXREAL_0:2; then
    z in {x: n <= #occurrences(x,A)} by A3;
    hence z in ROUGH(A,n) by A2,Def24;
  end;
