reserve x,x1,x2,x3 for Real;

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