reserve m, n for Nat;

theorem
  for p being Prime, n being Nat st 1 <= n & n * p is square-free
  holds Moebius (n * p) = - Moebius n
proof
  let p be Prime, a be Nat;
  assume that
A1: 1 <= a and
A2: a * p is square-free;
  a, p are_coprime
  proof
    assume not a, p are_coprime;
    then consider q being Prime such that
A4: q divides a and
A5: q divides p by PYTHTRIP:def 2;
    q = 1 or q = p by A5,INT_2:def 4;
    then p * p divides a * p by A4,INT_2:def 4,PYTHTRIP:7;
    then p |^ 2 divides a * p by WSIERP_1:1;
    hence thesis by A2;
  end;
  then Moebius (a * p) = (Moebius a) * (Moebius p) by A1,Th35
    .= (Moebius a) * (-1) by Th34;
  hence thesis;
end;
