
theorem Th8:
  for n,i be Nat st n >= 1 & i >= 1 holds (n + i)! >= n! + i
  proof
    let n,i be Nat;
    assume n >= 1 & i >= 1; then
    per cases by XXREAL_0:1;
    suppose n >= 1 & i > 1; then
      per cases by XXREAL_0:1;
      suppose n > 1 & i > 1;
        hence thesis by Th7;
      end;
      suppose
        n = 1 & i > 1;
        hence thesis by NEWTON:13,NEWTON:38;
      end;
    end;
    suppose n >= 1 & i = 1;
      hence thesis by Th4;
    end;
  end;
