theorem Th87:
  r <> 0 or n = 0 iff r|^n <> 0
proof
  defpred P[Nat] means r <> 0 or $1 = 0 iff r|^$1 <> 0;
A1: P[k] implies P[k+1]
  proof
    assume
A2: P[k];
    r|^(k+1) = r|^k*r by Th6;
    hence thesis by A2;
  end;
A3: P[0] by RVSUM_1:94;
  P[k] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
