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 Th38:
  m in dom (((Newton_Coeff n)|n)/^1) implies
    (((Newton_Coeff n)|n)/^1).m = (Newton_Coeff n).(m+1)
  proof
    A1: n = 0 implies ((Newton_Coeff n)|n) = {};
    n>0 implies n in dom (Newton_Coeff n) by Th30;
    hence thesis by A1,Th29;
  end;
