
theorem Th6:
  for p being Prime, n being non zero Nat holds p
  divides n iff p |-count n > 0
proof
  let p be Prime, n be non zero Nat;
A1: p <> 1 by INT_2:def 4;
  thus p divides n implies p |-count n > 0
  proof
    assume
A2: p divides n;
    p |-count n >= 1
    proof
      assume p |-count n < 1;
      then p |-count n = 0 by NAT_1:25;
      then not p |^(0+1) divides n by A1,NAT_3:def 7;
      hence contradiction by A2;
    end;
    hence thesis;
  end;
  assume p |-count n > 0;
  then reconsider d = p |-count n as non zero Nat;
  p <> 1 by INT_2:def 4;
  then p |^ d divides n by NAT_3:def 7;
  then p |^ (0+1) divides n by NAT_3:4;
  hence thesis;
end;
