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 Th35:
  for n be positive Nat holds
    Sum(Newton_Coeff n) = Sum(((Newton_Coeff n)|n)/^1) + 2
  proof
    let n be positive Nat;
    Sum(Newton_Coeff n) = Sum(((Newton_Coeff n)|n)/^1) +
      (Newton_Coeff n).1 + (Newton_Coeff n).(n+1) by Th34
    .= Sum(((Newton_Coeff n)|n)/^1) + (Newton_Coeff n).1 + 1 by Th32
    .= Sum(((Newton_Coeff n)|n)/^1) + 1 + 1 by Th33;
    hence thesis;
  end;
