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 Th54:
  for A being FinSequence, a being set holds #occurrences(a,A) <= len A
  proof
    let A be FinSequence;
    let a be set;
    {i: i in dom A & a in A.i} c= dom A
    proof
      let z be object; assume z in {i: i in dom A & a in A.i}; then
      ex i st z = i & i in dom A & a in A.i;
      hence thesis;
    end; then
    Segm #occurrences(a,A) c= Segm card dom A & dom A = Seg len A &
    card Seg len A = len A by CARD_1:11,FINSEQ_1:57,def 3;
    hence #occurrences(a,A) <= len A by NAT_1:39;
  end;
