
theorem
  for n being Nat st n <> 0 holds n < 2*n
proof
  let n be Nat;
  assume that
A1: n <> 0 and
A2: 2*n <= n;
  per cases by A2,XXREAL_0:1;
  suppose 2*n = n;
    hence contradiction by A1;
  end;
  suppose 2*n < n;
    then 2*n+-(1*n) < 1*n+-(1*n) by XREAL_1:6;
    hence contradiction by NAT_1:2;
  end;
end;
