reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem POWD:
  for n be non zero Nat, a be Integer, b be Integer holds
  b|^n divides a|^n iff b divides a
  proof
    let n be non zero Nat, a be Integer, b be Integer;
    thus b|^n divides a|^n implies b divides a
    proof
      assume
  A1: b|^n divides a|^n;
      |.b.||^n = |.b|^n.| by TAYLOR_2:1
      .= b|^n gcd a|^n by A1,NEWTON02:3
      .= (b gcd a)|^n by NEWTON027;
      hence thesis by NEWTON02:3, WSIERP_1:3;
    end;
    thus thesis by NEWTON02:15;
  end;
