reserve x,x1,x2,x3 for Real;

theorem Th40:
  sinh(x1)<>0 & sinh(x2)<>0 implies coth(x1+x2)=(1+coth(x1)*coth(
  x2))/(coth(x1)+coth(x2))
proof
  assume that
A1: sinh(x1)<>0 and
A2: sinh(x2)<>0;
A3: sinh.(x1)<>0 by A1,SIN_COS2:def 2;
A4: sinh.(x2)<>0 by A2,SIN_COS2:def 2;
  coth(x1+x2)=cosh.(x1+x2)/sinh(x1+x2) by SIN_COS2:def 4
    .=cosh.(x1+x2)/sinh.(x1+x2) by SIN_COS2:def 2
    .=((cosh.x1)*(cosh.x2) + (sinh.x1)*(sinh.x2)) /sinh.(x1+x2) by SIN_COS2:20
    .=((cosh.x1)*(cosh.x2) + (sinh.x1)*(sinh.x2)) /((sinh.x1)*(cosh.x2) + (
  cosh.x1)*(sinh.x2)) by SIN_COS2:21
    .=(((cosh.x1)*(cosh.x2) + (sinh.x1)*(sinh.x2))/((sinh.x1)*(sinh.x2))) /(
  ((sinh.x1)*(cosh.x2) + (cosh.x1)*(sinh.x2))/((sinh.x1)*(sinh.x2))) by A3,A4,
XCMPLX_1:6,55
    .=(((cosh.x1)*(cosh.x2)/((sinh.x1)*(sinh.x2))) + (sinh.x1)*(sinh.x2)/((
sinh.x1)*(sinh.x2))) /(((sinh.x1)*(cosh.x2) + (cosh.x1)*(sinh.x2))/((sinh.x1)*(
  sinh.x2))) by XCMPLX_1:62
    .=(((cosh.x1)*(cosh.x2)/((sinh.x1)*(sinh.x2)))+ 1) /(((sinh.x1)*(cosh.x2
  )+(cosh.x1)*(sinh.x2))/((sinh.x1)*(sinh.x2))) by A3,A4,XCMPLX_1:6,60
    .=(((cosh.x1)/(sinh.x1)*(cosh.x2)/(sinh.x2))+ 1) /(((sinh.x1)*(cosh.x2)+
  (cosh.x1)*(sinh.x2))/((sinh.x1)*(sinh.x2))) by XCMPLX_1:83
    .=(((cosh.x1)/(sinh.x1)*((cosh.x2)/(sinh.x2)))+ 1) /(((sinh.x1)*(cosh.x2
  )+(cosh.x1)*(sinh.x2))/((sinh.x1)*(sinh.x2))) by XCMPLX_1:74
    .=(((cosh.x1)/(sinh.x1))*((cosh.x2)/(sinh.x2))+ 1)/(((sinh.x1) *(cosh.x2
  ))/((sinh.x1)*(sinh.x2))+((cosh.x1)*(sinh.x2))/((sinh.x1) *(sinh.x2))) by
XCMPLX_1:62
    .=(((cosh.x1)/(sinh.x1))*((cosh.x2)/(sinh.x2))+ 1)/(((sinh.x1)/(sinh.x1)
*(cosh.x2))/(sinh.x2)+((cosh.x1)*(sinh.x2))/((sinh.x1) *(sinh.x2))) by
XCMPLX_1:83
    .=(((cosh.x1)/(sinh.x1))*((cosh.x2)/(sinh.x2))+ 1)/(((sinh.x1)/(sinh.x1)
*(cosh.x2))/(sinh.x2)+((cosh.x1)/(sinh.x1)*(sinh.x2))/( (sinh.x2))) by
XCMPLX_1:83
    .=(((cosh.x1)/(sinh.x1))*((cosh.x2)/(sinh.x2))+ 1)/(1* cosh.x2/sinh.x2+(
  (cosh.x1)/(sinh.x1)*(sinh.x2))/( (sinh.x2))) by A3,XCMPLX_1:60
    .=(((cosh.x1)/(sinh.x1))*((cosh.x2)/(sinh.x2))+ 1)/( cosh.x2/sinh.x2+
  cosh.x1/sinh.x1) by A4,XCMPLX_1:89
    .=(((cosh(x1))/(sinh.x1))*((cosh.x2)/(sinh.x2))+ 1)/( cosh.x2/sinh.x2+
  cosh.x1/sinh.x1) by SIN_COS2:def 4
    .=(((cosh(x1))/(sinh.x1))*((cosh(x2))/(sinh.x2))+ 1)/( cosh.x2/sinh.x2+
  cosh.x1/sinh.x1) by SIN_COS2:def 4
    .=(((cosh(x1))/(sinh.x1))*((cosh(x2))/(sinh.x2))+ 1)/( cosh(x2)/sinh.x2+
  cosh.x1/sinh.x1) by SIN_COS2:def 4
    .=(((cosh(x1))/(sinh.x1))*((cosh(x2))/(sinh.x2))+ 1)/( cosh(x2)/sinh.x2+
  cosh(x1)/sinh.x1) by SIN_COS2:def 4
    .=(((cosh(x1))/(sinh(x1)))*((cosh(x2))/(sinh.x2))+ 1)/( cosh(x2)/sinh.x2
  +cosh(x1)/sinh.x1) by SIN_COS2:def 2
    .=(((cosh(x1))/(sinh(x1)))*((cosh(x2))/(sinh.x2))+ 1)/( cosh(x2)/sinh(x2
  )+cosh(x1)/sinh.x1) by SIN_COS2:def 2
    .=(((cosh(x1))/(sinh(x1)))*((cosh(x2))/(sinh.x2))+ 1)/( cosh(x2)/sinh(x2
  )+cosh(x1)/sinh(x1)) by SIN_COS2:def 2
    .=(coth(x1)*coth(x2)+ 1)/(coth(x2)+coth(x1)) by SIN_COS2:def 2;
  hence thesis;
end;
