reserve x,x1,x2,x3 for Real;

theorem Th16:
  sin(3*x)=-4*(sin(x))|^3+3*sin(x)
proof
  sin(3*x)=sin((x+x)+x) .=sin(2*x)*cos(x)+cos(x+x)*sin(x) by SIN_COS:75
    .=(2*sin(x)*cos(x))*cos(x)+cos(2*x)*sin(x) by Th5
    .=2*sin(x)*(cos(x)*cos(x))+(1-2*(sin(x))^2)*sin(x) by Th7
    .=2*sin(x)*(1-(sin(x))^2)+(1-2*(sin(x))^2)*sin(x) by SIN_COS4:5
    .=2*sin(x)*1-2*sin(x)*(sin(x)*sin(x))+(1*sin(x)-2*(sin(x))^2*sin(x))
    .=2*sin(x)*1-2*((sin(x))|^1*(sin(x)))*sin(x) +(1*sin(x)-2*(sin(x))^2*sin
  (x))
    .=2*sin(x)*1-2*((sin(x))|^(1+1))*sin(x) +(1*sin(x)-2*(sin(x))^2*sin(x))
  by NEWTON:6
    .=2*sin(x)*1-2*((sin(x))|^2*sin(x)) +(1*sin(x)-2*(sin(x))^2*sin(x))
    .=2*sin(x)*1-2*(sin(x))|^(2+1) +(1*sin(x)-2*(sin(x))^2*sin(x)) by NEWTON:6
    .=2*sin(x)-2*(sin(x))|^3+(sin(x)-2* ((sin(x))|^1*sin(x))*sin(x))
    .=2*sin(x)-2*(sin(x))|^3+(sin(x)-2*((sin(x))|^(1+1))*sin(x)) by NEWTON:6
    .=2*sin(x)-2*(sin(x))|^3+(sin(x)-2*((sin(x))|^(1+1)*sin(x)))
    .=2*sin(x)-2*(sin(x))|^3+(sin(x)-2*((sin(x))|^(2+1))) by NEWTON:6
    .=-(-3*sin(x)+4*(sin(x))|^3);
  hence thesis;
end;
