
theorem Thm17:
  for x,y,z being Real st x + 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 such that
A1: x + y + z = PI;
    sin((x+y)-PI)=- sin(x+y) by COMPLEX2:5;
    then - sin((x+y)-PI)= sin(x+y); then
A2: (sin(-((x+y)-PI)))^2=(sin(x+y))^2 by SIN_COS:31;
A3: (cos y)^2 + (sin y)^2 = 1 by SIN_COS:29;
A4: (cos x)^2 + (sin x)^2 = 1 by SIN_COS:29;
A5: (sin(x+y))^2 = (sin x * cos y + cos x * sin y)^2 by SIN_COS:75
    .= (sin x * cos y)^2+ 2 * (sin x * cos y) * (cos x * sin y) +
    (cos x * sin y)^2 by SQUARE_1:4
    .= (sin x * cos y) * (sin x * cos y) +
    2 * sin x * cos y * cos x * sin y + (cos x * sin y)^2 by SQUARE_1:def 1
    .= (sin x * cos y) * (sin x * cos y) +
    2 * sin x * cos y * cos x * sin y + (cos x * sin y) * (cos x * sin y)
    by SQUARE_1:def 1
    .= (sin x * sin x) * (cos y * cos y) +
    2 * sin x * cos y * cos x * sin y + cos x * cos x * sin y * sin y
    .= (sin x)^2 * (cos y * cos y) + 2 * sin x * cos y * cos x * sin y +
    cos x * cos x * sin y * sin y by SQUARE_1:def 1
    .= (sin x)^2 * (cos y)^2 + 2 * sin x * cos y * cos x * sin y +
    (cos x * cos x) * sin y * sin y by SQUARE_1:def 1
    .= (sin x)^2 * (cos y)^2 + 2 * sin x * cos y * cos x * sin y +
    (cos x)^2 * sin y * sin y by SQUARE_1:def 1
    .= (sin x)^2 * (cos y)^2 + 2 * sin x * cos y * cos x * sin y +
    (cos x)^2 * (sin y * sin y)
    .= (sin x)^2 * (cos y)^2 + 2 * sin x * cos y * cos x * sin y +
    (cos x)^2 * (sin y)^2 by SQUARE_1:def 1;
    (sin x)^2+(sin y)^2 - 2 * sin x * sin y * cos z
    =(sin x)^2+(sin y)^2 - 2 * sin x * sin y * cos (- ((x+y)-PI) ) by A1
    .=(sin x)^2+(sin y)^2 - 2 * sin x * sin y * cos ((x+y)-PI) by SIN_COS:31
    .=(sin x)^2+(sin y)^2 - 2 * sin x * sin y * (- cos (x+y)) by COMPLEX2:5
    .=(sin x)^2+(sin y)^2 + 2 * sin x * sin y * cos (x+y)
    .=(sin x)^2+(sin y)^2 +
    2 * sin x * sin y * (cos x * cos y - sin x * sin y) by SIN_COS:75
    .=(sin x)^2+(sin y)^2 + 2 * sin x * sin y * cos x * cos y -
    2 * (sin x * sin x) * sin y * sin y
    .=(sin x)^2+(sin y)^2 + 2 * sin x * sin y * cos x * cos y -
    2 * (sin x)^2 * sin y * sin y by SQUARE_1:def 1
    .=(sin x)^2+(sin y)^2 + 2 * sin x * sin y * cos x * cos y -
    2 * (sin x)^2 * (sin y * sin y)
    .=(sin x)^2+(sin y)^2 + 2 * sin x * sin y * cos x * cos y -
    2 * (sin x)^2 * (sin y)^2 by SQUARE_1:def 1
    .=(sin x)^2 * (1 - (sin y)^2) +(sin y)^2 +
    2 * sin x * sin y * cos x * cos y - (sin x)^2 * (sin y)^2
    .=(sin x)^2 * (cos y)^2 +(sin y)^2 * ( 1 - (sin x)^2) +
    2 * sin x * sin y * cos x * cos y by A3
    .=(sin x)^2 * (cos y)^2 +(sin y)^2 * (cos x)^2 +
    2 * sin x * sin y * cos x * cos y by A4;
    hence thesis by A1,A2,A5;
  end;
