reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;

theorem Th40:
  j<>0 implies j divides (j+k)!
proof
  defpred P[Nat] means for j st j<>0 holds j divides (j+$1)!;
A1: for k st P[k] holds P[k+1]
  proof
    let k;
    assume
A2: for j st j<>0 holds j divides (j+k)!;
    let j;
    assume j<>0;
    then j divides ((j+k)!)*((j+k)+1) by A2,NAT_D:9;
    hence thesis by Th15;
  end;
A3: P[0] by Th35;
  for k holds P[k] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
