 reserve n,i for Nat;
 reserve p for Prime;

theorem DivRelPrime:
  for n being square-free non zero Nat,
      a being non zero Nat st a divides n holds
    a, n div a are_coprime
  proof
    let n be square-free non zero Nat,
        a be non zero Nat;
    assume
A0: a divides n;
    assume
Z1: not a, n div a are_coprime;
Z2: n div a <> 0
    proof
      assume n div a = 0; then
      n < a by NAT_2:12;
      hence thesis by NAT_D:7,A0;
    end;
    consider k being non zero Nat such that
A1: k <> 1 & k divides a & k divides n div a by RelPrimeEx,Z1,Z2;
    a * (n div a) = n by A0,NAT_D:3; then
    k ^2 divides n by A1,NAT_3:1;
    hence thesis by A1,SqCon1;
  end;
