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
  for a be non trivial Nat, p be prime Nat holds
    a divides p|^n implies p divides a
  proof
    let a be non trivial Nat, p be prime Nat;
    assume a divides p|^n; then
    A1: a gcd p|^n = |.a.| by NEWTON02:3;
    per cases by GCDP;
    suppose
      a gcd p = 1; then
      a gcd p|^n = 1 by WSIERP_1:12;
      hence thesis by A1,Def0;
    end;
    suppose a gcd p = p;
      hence thesis by INT_2:def 2;
    end;
  end;
