
theorem Th25:
  for n being non zero Element of NAT for x,y being Element of
  COMPLEX st x in n-roots_of_1 & y in n-roots_of_1 holds x*y in n-roots_of_1
proof
  let n be non zero Element of NAT;
  let x,y be Element of COMPLEX such that
A1: x in n-roots_of_1 and
A2: y in n-roots_of_1;
  reconsider a=x as Element of F_Complex by COMPLFLD:def 1;
  consider i being Element of NAT such that
A3: a = [** cos((2*PI*i)/n), sin((2*PI*i)/n) **] by A1,Th24;
  reconsider b=y as Element of F_Complex by COMPLFLD:def 1;
  consider j being Element of NAT such that
A4: b = [** cos((2*PI*j)/n), sin((2*PI*j)/n) **] by A2,Th24;
  a*b=[** cos((2*PI*((i+j) mod n))/n),sin((2*PI*((i+j)mod n))/n)**] by A3,A4
,Th11;
  hence thesis by Th24;
end;
