reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem Th36:
  k<=n implies (n!)/(k!) is integer
proof
  defpred P[Nat] means (($1+k)!)/(k!) is integer;
  assume k<=n;
  then reconsider m = n-k as Element of NAT by INT_1:5;
A1: n=m+k;
  now
    let m;
A2: (((m+1)+k)!)/(k!) = (m+k+1)*((m+k)!)*(k!)" by NEWTON:15
      .= (m+k+1)*(((m+k)!)*(k!)")
      .= (m+k+1)*(((m+k)!)/(k!));
    assume ((m+k)!)/(k!) is integer;
    then reconsider i = ((m+k)!)/(k!) as Integer;
    (m+k+1)*i is Integer;
    hence (((m+1)+k)!)/(k!) is integer by A2;
  end;
  then
A3: for n being Nat holds P[n] implies P[n+1];
A4: P[0] by XCMPLX_1:60;
  for n being Nat holds P[n] from NAT_1:sch 2(A4,A3);
  hence thesis by A1;
end;
