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

theorem ST:
  for a,b,c,d be Real, n be Nat holds
  (a*b,c*d) Subnomial n = ((a,d) Subnomial n) (#) ((b,c) Subnomial n)
    proof
    let a,b,c,d be Real, n be Nat;
    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
    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; then
    A1: dom ((a*b,c*d) Subnomial n) = dom ((a,d) Subnomial n) /\
      dom ((b,c) Subnomial n);
    for 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 by STT;
    hence thesis by A1,VALUED_1:def 4;
  end;
