reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem Th39:
  n is prime implies n divides (((Newton_Coeff n)|n)/^1).k
  proof
    L1: n is prime & k >= n implies n divides (((Newton_Coeff n)|n)/^1).k
    proof
      assume
      A1: n is prime & k >= n; then
      n in dom (Newton_Coeff n) by Th30; then
      k >= len ((Newton_Coeff n)|n) by A1,Th10; then
      k+1 > len ((Newton_Coeff n)|n) by NAT_1:13; then
      not k+1 in dom ((Newton_Coeff n)|n) by FINSEQ_3:25; then
      not k in dom (((Newton_Coeff n)|n)/^1) by FINSEQ_5:26; then
      (((Newton_Coeff n)|n)/^1).k = {} by FUNCT_1:def 2;
      hence thesis by NAT_D:6;
    end;
    n is prime & k < n implies n divides (((Newton_Coeff n)|n)/^1).k
    proof
      A0: k = (k+1) - 1;
      assume
      A1: n is prime & k < n;
      per cases;
      suppose
        A1aa: k > 0; then
        A1a: k+1 > 0+1 & k+1 < n+1 & len(Newton_Coeff n) = n+1
          by A1,XREAL_1:8,NEWTON:def 5; then
        A2: k+1 in dom (Newton_Coeff n) by FINSEQ_3:25;
        n in dom (Newton_Coeff n) by A1,Th30; then
        A3: len((Newton_Coeff n)|n) = n by Th10;
        k+1 <= n by A1,NAT_1:13; then
        (k)+1 in dom ((Newton_Coeff n)|n) by A1a,A3,FINSEQ_3:25; then
        k in dom (((Newton_Coeff n)|n)/^1) by A1aa,Th16; then
        (((Newton_Coeff n)|n)/^1).(k) = (Newton_Coeff n).(k+1) by Th38
        .= (n) choose (k) by A0,A2,NEWTON:def 5;
        hence thesis by A1,A1aa,Th21;
      end;
      suppose k = 0; then
        not k in dom (((Newton_Coeff n)|n)/^1) by FINSEQ_3:25; then
        (((Newton_Coeff n)|n)/^1).k = {} by FUNCT_1:def 2;
        hence thesis by NAT_D:6;
      end;
    end;
    hence thesis by L1;
  end;
