
theorem
  for x,y,z being Real st x - (-2*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 x - (-2*PI + y) + z = PI; then
A1: x + (- (-2*PI + y)) + z = PI;
    (sin x)^2 + (sin (-(-2*PI+y)))^2 - 2 * sin x * sin (-(-2*PI+y)) * cos z
    =(sin x)^2 + (sin (-(-2*PI+y)))^2 - 2 * sin x * (- sin (-2*PI+y)) *
    cos z by SIN_COS:31
    .=(sin x)^2 + (sin (-(-2*PI+y)))^2 + 2 * sin x * sin (-2*PI+y) * cos z
    .=(sin x)^2 + (sin (-2*PI+y))^2 + 2 * sin x * sin (-2*PI+y) * cos z
    by Thm16
    .=(sin x)^2 + (sin (y))^2 + 2 * sin x * sin (-2*PI+y) * cos z by Thm2
    .=(sin x)^2 + (sin (y))^2 + 2 * sin x * sin y * cos z by Thm2;
    hence thesis by A1,Thm17;
  end;
