reserve th, th1, th2, th3 for Real;

theorem
  cos(th1)<>0 & cos(th2)<>0 & cos(th3)<>0 implies tan(th1+th2+th3) = (
  tan(th1)+tan(th2)+tan(th3)-tan(th1)*tan(th2)*tan(th3)) /(1-tan(th2)*tan(th3)-
  tan(th3)*tan(th1)-tan(th1)*tan(th2))
proof
  assume that
A1: cos(th1)<>0 & cos(th2)<>0 and
A2: cos(th3)<>0;
A3: cos(th1)*cos(th2) <> 0 by A1;
  tan(th1+th2+th3) =cos(th1)*cos(th2)*cos(th3) *(tan(th1)+tan(th2)+tan(th3
  )-tan(th1)*tan(th2)*tan(th3))/cos(th1+th2+th3) by A1,A2,Th11
    .=cos(th1)*cos(th2)*cos(th3) *(tan(th1)+tan(th2)+tan(th3)-tan(th1)*tan(
th2)*tan(th3)) /(cos(th1)*cos(th2)*cos(th3) *(1-tan(th2)*tan(th3)-tan(th3)*tan(
  th1)-tan(th1)*tan(th2))) by A1,A2,Th12
    .=(cos(th1)*cos(th2)*cos(th3)/(cos(th1)*cos(th2)*cos(th3))) /((1-tan(th2
  )*tan(th3)-tan(th3)*tan(th1)-tan(th1)*tan(th2)) /(tan(th1)+tan(th2)+tan(th3)-
  tan(th1)*tan(th2)*tan(th3))) by XCMPLX_1:84
    .=1/((1-tan(th2)*tan(th3)-tan(th3)*tan(th1)-tan(th1)*tan(th2)) /(tan(th1
  )+tan(th2)+tan(th3)-tan(th1)*tan(th2)*tan(th3))) by A2,A3,XCMPLX_1:60
    .=(tan(th1)+tan(th2)+tan(th3)-(tan(th1)*tan(th2)*tan(th3))) /(1-tan(th2)
  *tan(th3)-tan(th3)*tan(th1)-tan(th1)*tan(th2)) by XCMPLX_1:57;
  hence thesis;
end;
