reserve a,b,i,j,k,l,m,n for Nat;

theorem SAA:
  for a be Real, n be Nat holds
    Sum ((a,a)Subnomial n) = (n+1)*a|^n
  proof
    let a be Real, n be Nat;
    A1: n+1 = len ((a,a)Subnomial (n+1-1));
    n+1 >= 0+1 by XREAL_1:6; then
    A2: n+1 in dom ((a,a)Subnomial n) by A1,FINSEQ_3:25;
    n+1 is set & n+1 in dom ((a,a)Subnomial n) &
      ((a,a) Subnomial n).(n+1) = a|^n by A2,CONST; then
    the_value_of ((a,a) Subnomial n) = a|^n by FUNCT_1:def 12;
    hence thesis by A1,RVSUM_3:7;
  end;
