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

theorem Th27:
  b <> 1 implies (a <> 0 & b |-count a = 0 iff not b divides a)
proof
  1 divides a by NAT_D:6;
  then
A1: b |^ 0 divides a by NEWTON:4;
  assume
A2: b <> 1;
  thus a <> 0 & b |-count a = 0 implies not b divides a
  proof
    assume that
A3: a <> 0 & b |-count a = 0 and
A4: b divides a;
    not b |^ (0+1) divides a by A2,A3,Def7;
    hence contradiction by A4;
  end;
  assume not b divides a;
  then ( not b |^ (0+1) divides a)& a <> 0 by NAT_D:6;
  hence thesis by A2,A1,Def7;
end;
