
theorem
  for L being Field, p,q being Polynomial of L, m being Element of NAT
  st m > 0 & len p <= m & len q <= m for x being Element of L st x
is_primitive_root_of_degree 2*m holds emb(2*m,L) <> 0.L implies (emb(2*m,L))" *
  DFT(DFT(p,x,2*m) * DFT(q,x,2*m), x", 2*m) = p *' q
proof
  let L be Field;
  let p,q be Polynomial of L;
  let m be Element of NAT;
  assume
A1: m > 0 & len p <= m & len q <= m;
  let x be Element of L;
  assume
A2: x is_primitive_root_of_degree 2*m;
  assume
A3: emb(2*m,L) <> 0.L;
  (emb(2*m,L))" * DFT(DFT(p,x,2*m) * DFT(q,x,2*m), x", 2*m) = (emb(2*m,L))
  " * DFT(DFT(p*'q,x,2*m), x", 2*m) by Th34
    .= (emb(2*m,L))" * (emb(2*m,L) * (p*'q)) by A1,A2,Th43
    .= ((emb(2*m,L))" * emb(2*m,L)) * (p*'q) by Th10
    .= 1.L * (p*'q) by A3,VECTSP_1:def 10
    .= p*'q by POLYNOM5:27;
  hence thesis;
end;
