
theorem Thm8:
  cos(PI/3) = 1/2
  proof
    set O=|[0,0]|;
    set A=|[0,1]|;
    set B=|[sqrt 3 /2,1/2]|;
    set a =|.B-O.|;
    set b =|.A-O.|;
    set c =|.A-B.|;
    |.O-A.|=|.O-B.| & |.O-A.|=|.A-B.| & |.O-A.|=1 by Lm3; then
A1: b=1 & c =1 & a=1 by EUCLID_6:43;
    1^2 = 1 * 1 by SQUARE_1:def 1
    .=1;
    then 1 = 1 + 1 - 2 * 1 * 1 * cos angle(B,O,A) by A1,EUCLID_6:7;
    hence thesis by Lm1,Lm6;
  end;
