reserve x,x1,x2,x3 for Real;

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