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 Th62:
  for A being FinSequence of bool X st A <> {} holds
  ROUGH(A, len A) = meet rng A
  proof
    let A be FinSequence of bool X such that
A1: A <> {};
    thus ROUGH(A, len A) c= meet rng A
    proof
      let z be object; assume
      z in ROUGH(A, len A); then
      z in {x: len A <= #occurrences(x,A)} by Def24; then
      consider x such that
A2:   z = x & len A <= #occurrences(x,A);
      #occurrences(x,A) <= len A by Th54;
      hence thesis by A1,A2,Th55,XXREAL_0:1;
    end;
    let z be object; assume
A3: z in meet rng A; then
    #occurrences(z,A) = len A by Th55; then
    z in {x: len A <= #occurrences(x,A)} by A3;
    hence thesis by A3,Def24;
  end;
