reserve x,x1,x2,x3 for Real;

theorem
  sinh(x1)<>0 & sinh(x2)<>0 implies coth(x1)+coth(x2)=sinh(x1+x2)/(sinh(
  x1)*sinh(x2)) & coth(x1)-coth(x2)=-(sinh(x1-x2))/(sinh(x1)*sinh(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;
A5: -(sinh(x1-x2))/(sinh(x1)*sinh(x2)) =-(sinh.(x1-x2))/(sinh(x1)*sinh(x2))
  by SIN_COS2:def 2
    .=-(sinh.(x1-x2))/(sinh.(x1)*sinh(x2)) by SIN_COS2:def 2
    .=-(sinh.(x1-x2))/(sinh.(x1)*sinh.(x2)) by SIN_COS2:def 2
    .=-((sinh.x1)*(cosh.x2)-(cosh.x1)*(sinh.x2))/(sinh.(x1)*sinh.(x2)) by
SIN_COS2:21
    .=-((sinh.x1)*(cosh.x2)/(sinh.(x1)*sinh.(x2)) -(cosh.x1)*(sinh.x2)/(sinh
  .(x1)*sinh.(x2))) by XCMPLX_1:120
    .=-((sinh.x1)/sinh.(x1)*(cosh.x2)/(sinh.(x2)) -(cosh.x1)*(sinh.x2)/(sinh
  .(x1)*sinh.(x2))) by XCMPLX_1:83
    .=-((sinh.x1)/sinh.(x1)*(cosh.x2)/(sinh.(x2)) -((sinh.x2)/sinh.(x2)*cosh
  .x1)/(sinh.(x1))) by XCMPLX_1:83
    .=-(1*(cosh.x2)/(sinh.(x2)) -((sinh.x2)/sinh.(x2)*cosh.x1)/(sinh.(x1)))
  by A3,XCMPLX_1:60
    .=-(cosh.x2/(sinh.(x2))-(1*cosh.x1)/(sinh.(x1))) by A4,XCMPLX_1:60
    .=-(cosh(x2)/(sinh.(x2))-(cosh.x1)/(sinh.(x1))) by SIN_COS2:def 4
    .=-(cosh(x2)/(sinh.(x2))-(cosh(x1))/(sinh.(x1))) by SIN_COS2:def 4
    .=-(cosh(x2)/(sinh(x2))-(cosh(x1))/(sinh.(x1))) by SIN_COS2:def 2
    .=-(coth(x2)-(cosh(x1))/(sinh(x1))) by SIN_COS2:def 2
    .=coth(x1)-coth(x2);
  sinh(x1+x2)/(sinh(x1)*sinh(x2)) =sinh.(x1+x2)/(sinh(x1)*sinh(x2)) by
SIN_COS2:def 2
    .=((sinh.x1)*(cosh.x2)+(cosh.x1)*(sinh.x2))/(sinh(x1)*sinh(x2)) by
SIN_COS2:21
    .=((sinh.x1)*(cosh.x2)+(cosh.x1)*(sinh.x2))/(sinh.(x1)*sinh(x2)) by
SIN_COS2:def 2
    .=((sinh.x1)*(cosh.x2)+(cosh.x1)*(sinh.x2))/(sinh.(x1)*sinh.(x2)) by
SIN_COS2:def 2
    .=(sinh.x1)*(cosh.x2)/(sinh.(x1)*sinh.(x2)) +(cosh.x1)*(sinh.x2)/(sinh.(
  x1)*sinh.(x2)) by XCMPLX_1:62
    .=(sinh.x1)/sinh.(x1)*(cosh.x2)/(sinh.(x2)) +(cosh.x1)*(sinh.x2)/(sinh.(
  x1)*sinh.(x2)) by XCMPLX_1:83
    .=1*(cosh.x2)/(sinh.(x2)) +(cosh.x1)*(sinh.x2)/(sinh.(x1)*sinh.(x2)) by A3,
XCMPLX_1:60
    .=(cosh.x2)/(sinh.(x2)) +(cosh.x1)/sinh.(x1)*(sinh.x2)/(sinh.(x2)) by
XCMPLX_1:83
    .=(cosh.x2)/(sinh.(x2)) +(cosh.x1)/sinh.(x1)*((sinh.x2)/(sinh.(x2))) by
XCMPLX_1:74
    .=(cosh.x2)/(sinh.(x2)) +(cosh.x1)/sinh.(x1)*1 by A4,XCMPLX_1:60
    .=(cosh.x2)/(sinh.(x2)) +(cosh.x1)/sinh(x1) by SIN_COS2:def 2
    .=(cosh.x2)/(sinh(x2)) +(cosh.x1)/sinh(x1) by SIN_COS2:def 2
    .=cosh(x2)/(sinh(x2)) +(cosh.x1)/sinh(x1) by SIN_COS2:def 4
    .=coth(x2)+coth(x1) by SIN_COS2:def 4;
  hence thesis by A5;
end;
