reserve x,x1,x2,x3 for Real;

theorem
  sin(x)<>0 implies cot(3*x)=((cot(x))|^3-3*cot(x))/(3*(cot(x))^2-1)
proof
  assume
A1: sin(x)<>0;
  cot(3*x)=cot(x+x+x)
    .=(cot(x)*cot(x)*cot(x)-cot(x)-cot(x)-cot(x)) /(cot(x)*cot(x)+cot(x)*cot
  (x)+cot(x)*cot(x)-1) by A1,SIN_COS4:14
    .=(cot(x)*cot(x)*cot(x)-3*cot(x))/(3*(cot(x))^2-1)
    .=((cot(x))|^1*cot(x)*cot(x)-3*cot(x))/(3*(cot(x))^2-1)
    .=((cot(x))|^(1+1)*cot(x)-3*cot(x))/(3*(cot(x))^2-1) by NEWTON:6
    .=((cot(x))|^(2+1)-3*cot(x))/(3*(cot(x))^2-1) by NEWTON:6;
  hence thesis;
end;
