
theorem
  for n, i being Nat st n >= 2 |^ (2 * i + 2) holds
    n / 2 >= (2 |^ i) * sqrt n
  proof
    let n, i be Nat;
    assume
A1: n >= 2 |^ (2 * i + 2);
    assume n / 2 < (2 |^ i) * sqrt n; then
    n / 2 * 2 < (2 |^ i) * sqrt n * 2 by XREAL_1:68; then
    n < 2 * (2 |^ i) * sqrt n; then
    n < (2 |^ (i + 1)) * sqrt n by NEWTON:6; then
    n ^2 < ((2 |^ (i + 1)) * sqrt n) ^2 by SQUARE_1:16; then
    n ^2 < (2 |^ (i + 1)) ^2 * (sqrt n) ^2; then
    n ^2 < (2 |^ (i + 1)) ^2 * n by SQUARE_1:def 2; then
    n ^2 < (2 |^ (i + 1)) |^2 * n by NEWTON:81; then
    n ^2 < (2 |^ (2 * (i + 1))) * n by NEWTON:9; then
    (n ^2) / n < (2 |^ (2 * i + 2)) * n / n by A1,XREAL_1:74; then
    n < (2 |^ (2 * i + 2)) * n / n by A1,XCMPLX_1:89;
    hence thesis by A1,XCMPLX_1:89;
  end;
