reserve X,Y,Z,x,y,y1,y2 for set,
  D for non empty set,
  k,n,n1,n2,m2,m1 for Nat,

  L,K,M,N for Cardinal,
  f,g for Function;
reserve r for Real;

theorem Th3:
  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
A2: r|^(k+1) = r|^k*r by NEWTON:6;
    assume P[k];
    hence thesis by A2;
  end;
A3: P[0] by NEWTON:4;
  P[k] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
