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
  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;
