
theorem Th1:
  for n being Nat st n > 1 holds n -' 1 <= 2 * n -' 3
proof
  let n be Nat;
  assume
A1: n > 1;
  then n -' 1 > 1 -' 1 by NAT_D:57;
  then
A2: n -' 1 + n > 0 + n by XREAL_1:6;
  2 * 1 < 2 * n by A1,XREAL_1:68;
  then 2 + 1 <= 2 * n by NAT_1:13;
  then
A3: 2 * n -' 3 = 2 * n - 3 by XREAL_1:233;
  n -' 1 = n - 1 by A1,XREAL_1:233;
  then 2 * n - 1 - 1 > n - 1 by A2,XREAL_1:9;
  then 2 * n - 2 - 1 >= n - 1 by INT_1:52;
  hence thesis by A1,A3,XREAL_1:233;
end;
