
theorem
zeta^2 = (-1/2) - <i> * (sqrt 3) / 2
proof
zeta = (-1/2) + <i> * ((sqrt 3) / 2); then
A: Re(zeta) = (-1/2) & Im(zeta) = (sqrt 3) / 2 by COMPLEX1:12; then
B: Re(zeta * zeta)
      = (-1/2)^2 - ((sqrt 3) / 2)^2 by COMPLEX1:16
     .= (-1/2)^2 - ((sqrt 3) / 2) * ((sqrt 3) / 2) by SQUARE_1:def 1
     .= (-1/2)^2 - ((sqrt 3) * (sqrt 3)) / (2 * 2)
     .= (-1/2)^2 - ((sqrt 3)^2) / (2 * 2) by SQUARE_1:def 1
     .= (-1/2)^2 - (3 / (2*2)) by SQUARE_1:def 2
     .= (-1/2) * (-1/2) - (3 / (2*2)) by SQUARE_1:def 1
     .= -1/2;
C: Im(zeta * zeta) = 2 * ((-1/2) * ((sqrt 3) / 2)) by A,COMPLEX1:16
                  .= -(sqrt 3) / 2;
Re((-1/2) + <i> * (-(sqrt 3) / 2)) = -1/2 &
Im((-1/2) + <i> * (-(sqrt 3) / 2)) = (-(sqrt 3) / 2) by COMPLEX1:12; then
zeta * zeta = (-1/2) + <i> * (-(sqrt 3) / 2) by B,C,COMPLEX1:3;
hence thesis by O_RING_1:def 1;
end;
