theorem
  for b be non trivial Nat, a be non zero Nat holds
    (b |-count a) = 0 iff a mod b <> 0
proof
  let b be non trivial Nat, a be non zero Nat;
  per cases;
  suppose
    A1: b |-count a <> 0; then
    b divides a by MOB16;
    hence thesis by A1,PEPIN:6;
  end;
  suppose
    A1: b |-count a = 0; then
    not b divides a by MOB16;
    hence thesis by A1,PEPIN:6;
  end;
end;
