reserve x,x1,x2,x3 for Real;

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