reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem
  for a st a <> 0 holds for m holds a GeoSeq.m <> 0
proof
  let a such that
A1: a <> 0;
  defpred P[Nat] means a GeoSeq.$1 <> 0;
A2: for n holds P[n] implies P[n+1]
  proof
    let n such that
A3: a GeoSeq.n <> 0;
    a GeoSeq.(n+1) = a GeoSeq.n * a by Th3;
    hence thesis by A1,A3;
  end;
A4: P[0] by Th3;
  thus for n holds P[n] from NAT_1:sch 2(A4,A2);
end;
