reserve x,x1,x2,x3 for Real;

theorem
  sinh x <> 0 implies (coth x)^2-(cosech x)^2=1
proof
  assume sinh(x)<>0;
  then
A1: (sinh x)^2<>0 by SQUARE_1:12;
  (coth x)^2-(cosech x)^2 =(cosh(x))^2/(sinh(x))^2-(1/sinh(x))^2 by XCMPLX_1:76
    .=(cosh(x))^2/(sinh(x))^2-1^2/(sinh(x))^2 by XCMPLX_1:76
    .=((cosh(x))^2-1)/(sinh(x))^2 by XCMPLX_1:120
    .=((cosh(x))^2-((cosh.x)^2-(sinh.x)^2))/(sinh(x))^2 by SIN_COS2:14
    .=((cosh(x))^2-(cosh.x)^2+(sinh.x)^2)/(sinh(x))^2
    .=((cosh(x))^2-(cosh(x))^2+(sinh.x)^2)/(sinh(x))^2 by SIN_COS2:def 4
    .=(0+(sinh(x))^2)/(sinh(x))^2 by SIN_COS2:def 2;
  hence thesis by A1,XCMPLX_1:60;
end;
