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 Th34:
  for n be positive Nat holds
  Sum(Newton_Coeff n) =
    Sum(((Newton_Coeff n)|n)/^1) + (Newton_Coeff n).1 + (Newton_Coeff n).(n+1)
  proof
    let n be positive Nat;
    len Newton_Coeff n = n+1 by NEWTON:def 5; then
    A2: Newton_Coeff n =((Newton_Coeff n)|n)^<*(Newton_Coeff n).(n+1)*>
      by RFINSEQ:7;
    A2a: n in dom Newton_Coeff n by Th30;
    A3: len ((Newton_Coeff n)|n) > 0;
    <*(Newton_Coeff n).(n+1)*> is FinSequence of REAL by RVSUM_1:145; then
    Sum(Newton_Coeff n) = Sum ((Newton_Coeff n)|n) +
      Sum <*(Newton_Coeff n).(n+1)*> by Th14,A2
    .= Sum ((Newton_Coeff n)|n) + (Newton_Coeff n).(n+1) by RVSUM_1:73
    .= (((Newton_Coeff n)|n).1 + Sum(((Newton_Coeff n)|n)/^1)) +
      (Newton_Coeff n).(n+1) by A3,IRRAT_1:17;
    hence thesis by A2a,Th9;
  end;
