
theorem
  for x,y,z being Real st (PI - x) - (PI - y) + z = PI holds
  (sin x)^2 + (sin y)^2 + 2 * sin x * sin y * cos z = (sin z)^2
  proof
    let x,y,z be Real;
    assume (PI - x) - (PI - y) + z = PI; then
A1: (PI - x) + (-(PI-y)) + z = PI;
    (sin (PI - x))^2 + (sin (-(PI - y)))^2 - 2 * sin (PI - x) *
    sin (-(PI-y)) * cos z
    =(sin (PI -x))^2 + (sin (-(PI - y)))^2 - 2 * sin (PI - x) *
    (- sin (PI- y)) * cos z by SIN_COS:31
    .=(sin (PI-x))^2 + (sin (-(PI-y)))^2 + 2 * sin (PI - x) * sin (PI - y)
    * cos z
    .=(sin (PI-x))^2 + (sin (PI-y))^2 + 2 * sin (PI-x) * sin (PI-y) * cos z
    by Thm16
    .=(sin x)^2 + (sin (PI-y))^2 + 2 * sin (PI-x) * sin (PI-y) * cos z
    by Thm1
    .=(sin x)^2 + (sin y)^2 + 2 * sin (PI-x) * sin (PI-y) * cos z by Thm1
    .=(sin x)^2 + (sin y)^2 + 2 * sin x * sin (PI-y) * cos z by Thm1
    .=(sin x)^2 + (sin y)^2 + 2 * sin x * sin y * cos z by Thm1;
    hence thesis by A1,Thm17;
  end;
