
theorem Th17:
  for R being left_zeroed right_zeroed add-cancelable
  add-associative distributive non empty doubleLoopStr, I being add-closed
  right-ideal non empty Subset of R, a being Element of I, n being Element of
  NAT holds n*a in I
proof
  let R be left_zeroed right_zeroed add-cancelable add-associative
distributive non empty doubleLoopStr, I be add-closed right-ideal non empty
  Subset of R, a be Element of I, n be Element of NAT;
  defpred P[Nat] means $1*a in I;
A1: for n being Nat holds P[n] implies P[n+1]
  proof
    let n be Nat;
A2: (n+1)*a = 1*a + n*a by BINOM:15
      .= a + n*a by BINOM:13;
    assume n*a in I;
    hence thesis by A2,Def1;
  end;
  0*a = 0.R by BINOM:12;
  then
A3: P[0] by Th3;
  for n being Nat holds P[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
