
theorem Th40:
  for L being Field, m being Element of NAT, x being Element of L
st m > 0 & x is_primitive_root_of_degree m holds VM(x,m) * VM(x",m) = VM(x",m)
  * VM(x,m)
proof
  let L be Field;
  let m be Element of NAT;
  let x be Element of L;
  assume that
 0 < m and
A1: x is_primitive_root_of_degree m;
  x <> 0.L by A1,Th30;
  then VM(x",m) * VM(x,m) = VM(x",m) * VM((x")",m) by VECTSP_1:24
    .= emb(m,L) * 1.(L,m) by A1,Th31,Th39;
  hence thesis by A1,Th39;
end;
