
theorem Th35:
  for S be non empty finite set,
  X be Subset of S,
  s,t be FinSequence of S,
  SD be Subset of dom s,
  x be set
  st SD = s"X & t = extract(s,SD) & x in X holds
  frequency(x,s) = frequency(x,t)
  proof
    let S be non empty finite set,
    X be Subset of S,
    s,t be FinSequence of S,
    SD be Subset of dom s,
    x be set;
    assume A1: SD = s"X &t = extract(s,SD) & x in X;then
    for a be object st a in {x} holds a in X by TARSKI:def 1;then
    {x} is Subset of X by TARSKI:def 3;
    hence thesis by Th34,A1;
  end;
