reserve x,x1,x2,x3 for Real;

theorem
  sinh(x/2)<>0 implies coth(x/2)=(sinh(x))/(cosh(x)-1)
proof
  assume sinh(x/2)<>0;
  then
A1: 2*sinh.(x/2)<>0 by SIN_COS2:def 2;
  (sinh(x))/(cosh(x)-1)=(sinh.(2*(x/2)))/(cosh(2*(x/2))-1) by SIN_COS2:def 2
    .=(sinh.(2*(x/2)))/(cosh.(2*(x/2))-1) by SIN_COS2:def 4
    .=(2*sinh.(x/2)*cosh.(x/2))/(cosh.(2*(x/2))-1) by SIN_COS2:23
    .=(2*sinh.(x/2)*cosh.(x/2))/(2*(cosh.(x/2))^2-1-1) by SIN_COS2:23
    .=(2*sinh.(x/2)*cosh.(x/2))/(2*((cosh.(x/2))^2-1))
    .=(2*sinh.(x/2)*cosh.(x/2))/(2*(sinh.(x/2))^2) by Lm3
    .=(2*sinh.(x/2)*cosh.(x/2))/(2*sinh.(x/2)*sinh.(x/2))
    .=cosh.(x/2)/sinh.(x/2) by A1,XCMPLX_1:91
    .=cosh(x/2)/sinh.(x/2) by SIN_COS2:def 4
    .=cosh(x/2)/sinh(x/2) by SIN_COS2:def 2;
  hence thesis;
end;
