reserve x,x1,x2,x3 for Real;

theorem
  cos(x)<>0 implies tan(3*x)=(3*tan(x)-(tan(x))|^3)/(1-3*(tan(x))^2)
proof
  assume
A1: cos(x)<>0;
  tan(3*x)=tan(x+x+x)
    .=(tan(x)+tan(x)+tan(x)-tan(x)*tan(x)*tan(x)) /(1-tan(x)*tan(x)-tan(x)*
  tan(x)-tan(x)*tan(x)) by A1,SIN_COS4:13
    .=(3*tan(x)-(tan(x))|^1*tan(x)*tan(x))/(1-3*tan(x)*tan(x))
    .=(3*tan(x)-(tan(x))|^(1+1)*tan(x))/(1-3*tan(x)*tan(x)) by NEWTON:6
    .=(3*tan(x)-(tan(x))|^(2+1))/(1-3*(tan(x)*tan(x))) by NEWTON:6;
  hence thesis;
end;
