
theorem
  for n being Nat st n > 1 holds (4 * n * n - 2*n) / (n + 1) > 1
proof
  defpred P[Nat] means (4 * $1 * $1 - 2*$1)/($1 + 1) > 1;
  let n be Nat;
A1: for k be non trivial Nat st P[k] holds P[k + 1]
  proof
    let k be non trivial Nat such that
A2: P[k];
    set k1 = k+1;
    (4 * k * k - 2*k) / (k + 1) = (4 * k * k - 2*k) * ( 1/ (k + 1)) by
XCMPLX_1:99;
    then (4 * k * k - 2*k) * (1 / (k + 1)) * (k + 1) > 1 * (k + 1) by A2,
XREAL_1:68;
    then (4 * k * k - 2*k) > 1 * (k + 1) by XCMPLX_1:109;
    then (4 * k * k - 2*k)-(k + 1) > 0 by XREAL_1:50;
    then (4 * k * k - 3*k - 1) + (8 * k + 1) > 0 + 0;
    then (4 * k1 * k1 - 2*k1 - (k1 + 1)) + (k1 + 1) > 0 + (k1 + 1) by XREAL_1:8
;
    then (4 * k1 * k1 - 2 * k1) / (k1 + 1) > (k1 + 1) / (k1 + 1) by XREAL_1:74;
    hence thesis by XCMPLX_1:60;
  end;
  assume n > 1;
  then
A3: n is non trivial by NAT_2:28;
A4: P[2];
  for k be non trivial Nat holds P[k] from NAT_2:sch 2 (A4, A1);
  hence thesis by A3;
end;
