
theorem ::: WSIERP_1:11
  for a,b be Integer, m,n be non zero Nat holds
    a,b are_coprime iff a|^m, b|^n are_coprime
  proof
    let a,b be Integer, m,n be non zero Nat;
    L1: a,b are_coprime implies a|^m,b|^n are_coprime
    proof
      assume a,b are_coprime; then
      a,b|^n are_coprime by WSIERP_1:10;
      hence thesis by WSIERP_1:10;
    end;
    a|^m,b|^n are_coprime implies a,b are_coprime
    proof
      assume
      A1: a|^m,b|^n are_coprime;
      reconsider k = m - 1 as Nat;
      reconsider l = n - 1 as Nat;
      a|^(k+1) = a*a|^k & b|^(l + 1) = b*b|^l by NEWTON:6;
      hence thesis by A1,N0141;
    end;
    hence thesis by L1;
  end;
