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

theorem
  for a,b be Real, n be Nat holds
    (a,b) Subnomial (n+2) = <*a|^(n+2)*>^(a*b*(a,b) Subnomial n)^<*b|^(n+2)*>
proof
  let a,b be Real, n be Nat;
  reconsider f = (b,a) Subnomial n as complex-valued FinSequence;
A0:  Rev ((b,a) Subnomial (n+1)) = Rev (<*b|^(n+1)*>^(a*(b,a) Subnomial n))
    by SAN
  .= (Rev(a*(b,a) Subnomial n))^(Rev<*b|^(n+1)*>) by FINSEQ_5:64
  .= (a*(Rev(b,a) Subnomial n))^(Rev<*b|^(n+1)*>) by CREV
  .= (a*((a,b)Subnomial n))^<*b|^(n+1)*> by SAB;
  (a,b) Subnomial (n+1+1) = <*a|^(n+2)*>^(b*(a,b) Subnomial (n+1)) by SAN
  .= <*a|^(n+2)*>^(b*Rev((b,a) Subnomial (n+1))) by SAB
  .= <*a|^(n+2)*>^((b*(a*(a,b)Subnomial n))^(b*<*b|^(n+1)*>)) by A0,ADP
  .= <*a|^(n+2)*>^(b*(a*(a,b)Subnomial n))^(b*<*b|^(n+1)*>) by FINSEQ_1:32
  .= <*a|^(n+2)*>^(b*(a*(a,b)Subnomial n))^(<*b*b|^(n+1)*>) by AMP
  .= <*a|^(n+2)*>^(b*(a*(a,b)Subnomial n))^(<*b|^((n+1)+1)*>) by NEWTON:6
  .= <*a|^(n+2)*>^(b*a*(a,b)Subnomial n)^(<*b|^((n+1)+1)*>) by VALUED_2:16;
  hence thesis;
end;
