reserve x,x1,x2,x3 for Real;

theorem
  sinh(x)<>0 implies coth(2*x)=(1+(coth(x))^2)/(2*coth(x))
proof
  assume sinh(x)<>0;
  then
A1: sinh.x<>0 by SIN_COS2:def 2;
  then
A2: (sinh.(x))^2 <>0 by SQUARE_1:12;
  coth(2*x)=cosh.(2*x)/sinh(2*x) by SIN_COS2:def 4
    .=cosh.(2*x)/sinh.(2*x) by SIN_COS2:def 2
    .=(2*(cosh.x)^2 - 1)/sinh.(2*x) by SIN_COS2:23
    .=(2*(cosh.x)^2 - 1)/(2*(sinh.x)*(cosh.x)) by SIN_COS2:23
    .=((2*(cosh.x)^2 - 1)/(sinh.(x))^2)/((2*(sinh.x)*(cosh.x))/ (sinh.(x))^2
  ) by A2,XCMPLX_1:55
    .=((2*(cosh.x)^2-((cosh.x)^2-(sinh.x)^2))/(sinh.(x))^2)/((2* (sinh.x)*(
  cosh.x))/(sinh.(x))^2) by SIN_COS2:14
    .=(((2-1)*(cosh.x)^2+(sinh.x)^2)/(sinh.(x))^2)/((2* (sinh.x)*(cosh.x))/(
  sinh.(x))^2)
    .=((cosh.x)^2/(sinh.(x))^2+(sinh.x)^2/(sinh.(x))^2)/((2* (sinh.x)*(cosh.
  x))/(sinh.(x))^2) by XCMPLX_1:62
    .=((cosh.x/sinh.x)^2+(sinh.x)^2/(sinh.(x))^2)/((2* (sinh.x)*(cosh.x))/(
  sinh.(x))^2) by XCMPLX_1:76
    .=((cosh.x/sinh.x)^2+(sinh.x/sinh.x)^2)/((2* (sinh.x)*(cosh.x))/(sinh.(x
  ))^2) by XCMPLX_1:76
    .=((cosh.x/sinh.x)^2+1^2)/(2*cosh.x*sinh.x/(sinh.x*sinh.x)) by A1,
XCMPLX_1:60
    .=((cosh.x/sinh.x)^2+1)/(2*cosh.x*sinh.x/sinh.x/sinh.x) by XCMPLX_1:78
    .=((cosh.x/sinh.x)^2+1)/(2*cosh.x/sinh.x) by A1,XCMPLX_1:89
    .=((cosh.x/sinh.x)^2+1)/(2*(cosh.x/sinh.x)) by XCMPLX_1:74
    .=((cosh(x)/sinh.x)^2+1)/(2*(cosh.(x)/sinh.x)) by SIN_COS2:def 4
    .=((cosh(x)/sinh.x)^2+1)/(2*(cosh(x)/sinh.x)) by SIN_COS2:def 4
    .=((cosh(x)/sinh(x))^2+1)/(2*(cosh(x)/sinh.x)) by SIN_COS2:def 2
    .=((coth(x))^2+1)/(2*(cosh(x)/sinh(x))) by SIN_COS2:def 2;
  hence thesis;
end;
