reserve th, th1, th2, th3 for Real;

theorem
  sin((th1-th2)/2)<>0 implies (sin(th1)-sin(th2))/(cos(th2)-cos(th1)) =
  cot((th1+th2)/2)
proof
  assume
A1: sin((th1-th2)/2)<>0;
  (sin(th1)-sin(th2))/(cos(th2)-cos(th1)) = 2*(cos((th1+th2)/2)*sin((th1-
  th2)/2))/(cos(th2)-cos(th1)) by Th16
    .= 2*(cos((th1+th2)/2)*sin((th1-th2)/2)) /(-2*(sin((th2+th1)/2)*sin((th2
  -th1)/2))) by Th18
    .= 2*(cos((th1+th2)/2)*sin((th1-th2)/2)) /(2*((sin((th2+th1)/2)*(-sin((
  th2-th1)/2)))))
    .= 2*(cos((th1+th2)/2)*sin((th1-th2)/2)) /(2*((sin((th2+th1)/2)*sin(-((
  th2-th1)/2))))) by SIN_COS:31
    .= (2/2)*(cos((th1+th2)/2)*sin((th1-th2)/2) /(sin((th2+th1)/2)*sin(((th1
  -th2))/2))) by XCMPLX_1:76
    .= (cos((th1+th2)/2)/(sin((th2+th1)/2)) *(sin((th1-th2)/2)/sin((th1-th2)
  /2))) by XCMPLX_1:76
    .= cot((th1+th2)/2) by A1,XCMPLX_1:88;
  hence thesis;
end;
