reserve m, n for Nat;

theorem Th27:
  for p being Prime, m, d being Nat st m is square-free
  & p divides m & d divides (m div p) holds d divides m & not p divides d
proof
  let p be Prime, m, d be Nat;
  assume that
A1: m is square-free and
A2: p divides m;
  assume d divides (m div p);
  then consider z being Nat such that
A3: m div p = d * z by NAT_D:def 3;
A4: (m div p) * p = d * z * p by A3;
  then m = d * (z * p) by A2,NAT_D:3;
  hence d divides m by NAT_D:def 3;
  assume p divides d;
  then consider w being Nat such that
A5: d = p * w by NAT_D:def 3;
  m = w * (p * p) * z by A2,A4,A5,NAT_D:3
    .= w * (p |^ 2) * z by WSIERP_1:1
    .= (p |^ 2) * (w * z);
  then p |^ 2 divides m by NAT_D:def 3;
  hence thesis by A1;
end;
