reserve m, n for Nat;

theorem
  for n being non zero Nat holds
  { k where k is Element of NAT : 0 < k & k divides n & k is square-free } =
  { k where k is Element of NAT : 0 < k & k divides Radical n }
proof
  let n be non zero Nat;
  thus { k where k is Element of NAT : 0 < k & k divides n & k is square-free
  } c= { k where k is Element of NAT : 0 < k & k divides Radical n }
  proof
    let x be object;
    assume x in { k where k is Element of NAT : 0 < k & k divides n & k is
    square-free };
    then consider k1 being Element of NAT such that
A1: k1 = x and
A2: 0 < k1 and
A3: k1 divides n & k1 is square-free;
    k1 divides Radical n by A2,A3,Th65;
    hence thesis by A1,A2;
  end;
  let x be object;
  assume x in { k where k is Element of NAT : 0 < k & k divides Radical n };
  then consider k1 being Element of NAT such that
A4: x = k1 & 0 < k1 and
A5: k1 divides Radical n;
A6: k1 is square-free by A5,Th26;
  Radical n divides n by Th55;
  then k1 divides n by A5,NAT_D:4;
  hence thesis by A4,A6;
end;
