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

theorem LS:
  for a,b be Real holds
    (a,b) Subnomial (n+1) = <*a|^(n+1)*>^(b*((a,b)Subnomial n))
proof
  let a,b be Real;
  A2: len ((a,b) Subnomial ((n+1)+1-1)) = (n+1) + 1 &
  len ((a,b) Subnomial (n+1-1)) = n+1; then
  A3: len ((a,b) Subnomial (n+1))
  = len <*a|^(n+1)*> + len ((a,b) Subnomial n) by FINSEQ_1:39
  .= len <*a|^(n+1)*> + len (b*((a,b)Subnomial n )) by NEWTON:2
  .= len (<*a|^(n+1)*>^(b*((a,b)Subnomial n))) by FINSEQ_1:22;
 for k be Nat st 1 <= k <= len ((a,b) Subnomial (n+1)) holds
 ((a,b) Subnomial (n+1)).k = (<*a|^(n+1)*>^(b*((a,b)Subnomial n))).k
 proof
   let k be Nat such that
   B0: 1 <= k <= len ((a,b) Subnomial (n+1));
   per cases by B0,XXREAL_0:1;
   suppose
     k > 1; then
     k >= 1+1 by NAT_1:13; then
     reconsider m = k-2 as Element of NAT by NAT_1:21;
     C2: n+2 >= m+2 by A2,B0; then
     reconsider l = n - m as Element of NAT by XREAL_1:6,NAT_1:21;
     C3: l = (n+1) - (m+1);
     m <= n by C2,XREAL_1:6; then
     C3a: 0+1 <= m+1 <= n+1 by XREAL_1:6; then
     C4: m+1 in dom (a,b) Subnomial n by A2,FINSEQ_3:25;
     C5: m = (m+1)-1 & l = n - m;
     C6:  1 <= m+1 <= len (b*((a,b)Subnomial n)) by C3a,A2,NEWTON:2;
     m + 1 = k - 1 & k in dom ((a,b) Subnomial (n+1)) by B0,FINSEQ_3:25; then
     ((a,b) Subnomial (n+1)).k = a|^l*b|^(m+1) by Def2,C3
     .= a|^l*(b*b|^m) by NEWTON:6
     .= b*(a|^l*b|^m)
     .= b*((a,b) Subnomial n).(m+1) by C4,C5,Def2
     .= (b*(a,b) Subnomial n).(m+1) by RVSUM_1:45
     .=(<*a|^(n+1)*>^(b*((a,b)Subnomial n))).(len (<*a|^(n+1)*>)+(m+1))
       by C6,FINSEQ_1:65
     .=(<*a|^(n+1)*>^(b*((a,b)Subnomial n))).((m+1)+1) by FINSEQ_1:39;
     hence thesis;
   end;
   suppose
     C1: k = 1;
     (<*a|^(n+1)*>^(b*(a,b) Subnomial n)).1
     = ((a,b) In_Power (n+1)).1 by NEWTON:28;
     hence thesis by NS,C1;
   end;
 end;
  hence thesis by A3;
end;
