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

theorem STT:
  for a,b,c,d be Real, n be Nat, x be object st
    x in dom ((a*b,c*d) Subnomial n) holds
  ((a*b,c*d) Subnomial n).x = ((a,d) Subnomial n).x * ((b,c) Subnomial n).x
  proof
    let a,b,c,d be Real, n be Nat, x be object such that
    B1: x in dom ((a*b,c*d) Subnomial n);
    len ((a*b,c*d) Subnomial (n+1-1)) = len ((a,d) Subnomial (n+1-1)) &
    len ((a*b,c*d) Subnomial (n+1-1)) = len ((b,c) Subnomial (n+1-1)); then
    A0: dom ((a*b,c*d) Subnomial n) = dom ((a,d) Subnomial n) &
    dom ((a*b,c*d) Subnomial n) = dom ((b,c) Subnomial n) by FINSEQ_3:29;
    reconsider x as positive Nat by B1,FINSEQ_3:25;
    set m = x-1;
    len ((a*b,c*d) Subnomial (n+1-1)) >= x by B1,FINSEQ_3:25; then
    consider k be Nat such that
    B2: n+1 = x + k by NAT_1:10;
    B3: n = m + k & k = n - m by B2; then
    ((a*b,c*d) Subnomial (m+k)).(m+1) = (a*b)|^k * (c*d)|^m by B1,Def2
    .= a|^k * b|^k * (c*d)|^m by NEWTON:7
    .= a|^k * b|^k * (c|^m * d|^m) by NEWTON:7
    .= (a|^k * d|^m)*(b|^k * c|^m)
    .= ((a,d) Subnomial (m+k)).(m+1)*(b|^k * c|^m) by A0,B1,B3,Def2
    .= ((a,d) Subnomial (m+k)).(m+1)*((b,c)Subnomial (m+k)).(m+1)
      by A0,B1,B3,Def2;
    hence thesis by B2;
  end;
