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

theorem CONST:
  for a be Real, n,i be Nat st i in dom ((a,a)Subnomial n) holds
  ((a,a)Subnomial n).i = a|^n
  proof
    let a be Real, n,i be Nat such that
    A1: i in dom ((a,a)Subnomial n);
    A2: 1 <= i <= len ((a,a) Subnomial (n+1-1)) by A1,FINSEQ_3:25; then
    reconsider k = i-1 as Nat;
    n+1 >= k+1 by A2; then
    reconsider l = n-k as Element of NAT by XREAL_1:6,NAT_1:21;
    a|^k*a|^l = a|^(k+l) by NEWTON:8;
    hence thesis by A1,Def2;
  end;
