reserve a, b, n for Nat,
  r for Real,
  f for FinSequence of REAL;
reserve p for Prime;

theorem Th30:
  for a, b being non zero Nat st b divides a holds p |-count b <= p |-count a
proof
  let a,b be non zero Nat;
  set x = p |-count a;
  set y = p |-count b;
  set z = p |-count (a div b);
  0+1 <= p |^ z by NAT_1:13;
  then
A1: 1 * p |^ y <= p |^ z * p |^ y by XREAL_1:66;
  assume b divides a;
  then
A2: a = b * (a div b) by NAT_D:3;
  then a div b <> 0;
  then p > 1 & p |^ y <= p |^ x by A2,A1,Th29,INT_2:def 4;
  hence thesis by PEPIN:66;
end;
