reserve m, n for Nat;

theorem
  for n being non zero Nat, A being finite Subset of NAT st
  A =
   {k where k is Element of NAT: 0 < k & k divides n & k is square-containing}
  holds SMoebius A = EmptyBag NAT
proof
  let n be non zero Nat, A be finite Subset of NAT;
  assume
A1: A = { k where k is Element of NAT : 0 < k & k divides n & k is
  square-containing };
A2: A misses SCNAT
  proof
    assume A meets SCNAT;
    then consider x being object such that
A3: x in A and
A4: x in SCNAT by XBOOLE_0:3;
    ex k being Element of NAT st k = x & 0 < k & k divides n & k is
    square-containing by A1,A3;
    hence thesis by A4,Def2;
  end;
  for x being object st x in NAT holds (SMoebius A).x = (EmptyBag NAT).x
  proof
    let x be object;
    assume x in NAT;
    then reconsider k = x as Element of NAT;
    support SMoebius A = A /\ SCNAT by Def5
      .= {} by A2;
    then (SMoebius A).k = 0 by PRE_POLY:def 7;
    hence thesis by PBOOLE:5;
  end;
  hence thesis by PBOOLE:3;
end;
