reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;
reserve p,q for natural Number;
reserve i0,i,i1,i2,i4 for Integer;
reserve x for set;
reserve p for Prime;
reserve i for Nat;
reserve x for Real;
reserve k for Nat;
reserve k,n,n1,n2,m1,m2 for Nat;

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;
