reserve x,x1,x2,x3 for Real;

theorem
  (sin(x))|^4=(3-4*cos(2*x)+cos(4*x))/8
proof
  (3-4*cos(2*x)+cos(2*2*x))/8 =(3-4*cos(2*x)+cos(2*(2*x)))/8
    .=(3-4*cos(2*x)+(1-2*(sin(2*x))^2))/8 by Th7
    .=(3-4*cos(2*x)+(1-2*(2*sin(x)*cos(x))^2))/8 by Th5
    .=(3-4*(1-2*(sin(x))^2)+(1-8*(sin(x))^2*(cos(x))^2))/8 by Th7
    .=(sin(x))^2*(1-cos(x)*cos(x))
    .=sin(x)*sin(x)*(sin(x)*sin(x)) by SIN_COS4:4
    .=(sin(x))|^1*sin(x)*(sin(x)*sin(x))
    .=(sin(x))|^(1+1)*(sin(x)*sin(x)) by NEWTON:6
    .=(sin(x))|^(1+1)*sin(x)*sin(x)
    .=(sin(x))|^(2+1)*sin(x) by NEWTON:6
    .=(sin(x))|^(3+1) by NEWTON:6;
  hence thesis;
end;
