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
  Sum(Newton_Coeff n) = Sum ((Newton_Coeff n)|n) + 1
  proof
    A0: Newton_Coeff n is FinSequence of NAT by Th1;
    A1: (Newton_Coeff n).(n+1) = ((1,1)In_Power n).(n+1) by NEWTON:31
    .= 1^n by NEWTON:29;
    A2: n+1 = len (Newton_Coeff n) by NEWTON:def 5;
    n+1 >= 0+1 by XREAL_1:6; then
    A3: n+1 in dom (Newton_Coeff n) by FINSEQ_3:25,A2;
    (Newton_Coeff n)
    = ((Newton_Coeff n)|n)^<*(Newton_Coeff n)/.(n+1)*> by A0,A2,FINSEQ_5:21
    .= ((Newton_Coeff n)|n)^<*(Newton_Coeff n).(n+1)*> by A3,PARTFUN1:def 6;
    hence thesis by A1,RVSUM_1:74;
  end;
