reserve m, n for Nat;

theorem Th17:
  for m, n being non zero Nat st (for p being Prime
  holds p |-count m <= p |-count n) holds support ppf m c= support ppf n
proof
  let m, n be non zero Nat;
  assume
A1: for p being Prime holds p |-count m <= p |-count n;
  let x be object;
  assume
A2: x in support ppf m;
  then x in support pfexp m by NAT_3:def 9;
  then reconsider p = x as Prime by NAT_3:34;
  (ppf m).p <> 0 by A2,PRE_POLY:def 7;
  then p |-count m <> 0 by NAT_3:55;
  then p |-count n > 0 by A1;
  then (ppf n).p = p |^ (p |-count n) by NAT_3:56;
  hence thesis by PRE_POLY:def 7;
end;
