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

theorem SAB:
  for a,b be Real, n be Nat holds
    (a,b) Subnomial n = Rev ((b,a) Subnomial n)
  proof
    let a,b be Real, n be Nat;
    A1: dom (a,b) Subnomial n = dom (Newton_Coeff n) & dom (b,a) Subnomial n =
    dom (Newton_Coeff n) by DOMN; then
    A2: dom ((a,b) Subnomial n) = dom (Rev ((b,a) Subnomial n))
      by FINSEQ_5:57;
    for i be object st i in dom ((a,b)Subnomial n) holds
    ((a,b)Subnomial n).i = (Rev((b,a)Subnomial n)).i
    proof
      let i be object such that
      B1: i in dom ((a,b)Subnomial n);
      reconsider i as Nat by B1;
      reconsider m = i-1 as Nat by B1,FINSEQ_3:26;
      len((a,b)Subnomial(n+1-1)) - (m+1) is Element of NAT
        by B1,FINSEQ_3:26; then
      reconsider l = n-m as Nat;
      set k = l+1;
      B2: i in dom Rev((b,a) Subnomial (n+1-1)) by A1,FINSEQ_5:57,B1;
      k+i = (len((b,a) Subnomial (n+1-1)))+1
      .= (len (Rev((b,a) Subnomial (n+1-1))))+1 by FINSEQ_5:def 3; then
      B3: k in dom Rev(Rev((b,a) Subnomial (n+1-1))) by B2,FINSEQ_5:59;
      B4: l = k-1;
      B5: m = n-l;
      ((a,b)Subnomial n).i = a|^l * b|^m by B1,Def2
      .= ((b,a)Subnomial n).(len ((b,a)Subnomial (n+1-1))-i+1)
        by B3,B4,B5,Def2
      .= Rev((b,a)Subnomial n).i by FINSEQ_5:58,A1,B1;
      hence thesis;
    end;
    hence thesis by A2,FUNCT_1:2;
  end;
