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 NAT330:
  for a,b,c be Nat st a <> 1 & b <> 0 & c <> 0 &
    b > a |-count c holds not a|^b divides c
  proof
    let a,b,c be Nat;
    assume a <> 1 & b <> 0 & c <> 0; then
    A2: a|^(a|-count c) divides c & not a|^((a|-count c)+1) divides c
      by NAT_3:def 7;
    assume b > a |-count c; then
    b >= a |-count c + 1 by NAT_1:13; then
    consider k be Nat such that
    A3: b = (a|-count c + 1) + k by NAT_1:10;
    a|^b = a|^(a|-count c + 1)*a|^k by A3,NEWTON:8;
    hence thesis by A2,INT29;
  end;
