
theorem
  for n,i being Nat st n >= 2 & i >= 1 holds (n+i)! > n! + i
  proof
    let n,i be Nat;
    assume n >= 2 & i >= 1; then
    per cases by XXREAL_0:1;
    suppose n >= 2 & i > 1; then
      per cases by XXREAL_0:1;
      suppose
A1:     n > 2 & i > 1; then
A2:     i >= 1+1 by NAT_1:13;
A3:     (n+i)! >= n! * (i!) by NEWTON04:70;
        n! > 1 & i! > 1 by A2,A1,ASYMPT_1:55; then
A5:     n! * (i!) >= n! + (i!) by Th1;
        per cases by A2,XXREAL_0:1;
        suppose
A6:       i = 2;
          n > 1 by A1,XXREAL_0:2;
          hence thesis by A6,Th6;
        end;
        suppose
          i > 2; then
A7:       i >= 2 + 1 by NAT_1:13;
          i! > i by ASYMPT_1:59,A7; then
          n! + (i!) > n! + i by XREAL_1:6; then
          n! * (i!) > n! + i by A5,XXREAL_0:2;
          hence thesis by A3,XXREAL_0:2;
        end;
      end;
      suppose
A8:      n = 2 & i > 1; then
         2 + i >= 2 + 1 by XREAL_1:7;
        hence thesis by A8,NEWTON:14,ASYMPT_1:59;
      end;
    end;
    suppose n >= 2 & i = 1;
    hence thesis by Th5;
    end;
  end;
