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;
