reserve x,x1,x2,x3 for Real;

theorem
  cos(x/2)<>0 implies cot(x/2)=(1+cos(x))/sin(x)
proof
  assume cos(x/2)<>0;
  then
A1: 2*cos(x/2)<>0;
  (1+cos(x))/sin(x)=(1+(2*(cos(x/2))^2-1))/sin(2*(x/2)) by Th7
    .=(2*(cos(x/2)*cos(x/2)))/(2*sin(x/2)*cos(x/2)) by Th5
    .=(2*cos(x/2)*cos(x/2))/(2*cos(x/2)*sin(x/2))
    .=cot(x/2) by A1,XCMPLX_1:91;
  hence thesis;
end;
