reserve x,x1,x2,x3 for Real;

theorem
  exp_R(x)-exp_R(-x)<>0 implies tanh(x)*coth(x)=1
proof
  assume
A1: exp_R(x)-exp_R(-x)<>0;
  exp_R(x)>0 by SIN_COS:55;
  then
A2: exp_R(x) + exp_R(-x)>0+0 by SIN_COS:55,XREAL_1:8;
  tanh(x)*coth(x)=tanh.x*coth(x) by SIN_COS2:def 6
    .=(exp_R.x-exp_R.(-x))/(exp_R.x+exp_R.(-x))*coth(x) by SIN_COS2:def 5
    .=(exp_R.x-exp_R.(-x))/(exp_R.x+exp_R.(-x))* ((exp_R(x)+exp_R(-x))/(
  exp_R(x)-exp_R(-x))) by Th36
    .=(exp_R.x-exp_R.(-x))/(exp_R.x+exp_R(-x))* ((exp_R(x)+exp_R(-x))/(exp_R
  (x)-exp_R(-x))) by SIN_COS:def 23
    .=(exp_R(x)-exp_R.(-x))/(exp_R.x+exp_R(-x))* ((exp_R(x)+exp_R(-x))/(
  exp_R(x)-exp_R(-x))) by SIN_COS:def 23
    .=(exp_R(x)-exp_R(-x))/(exp_R.x+exp_R(-x))* ((exp_R(x)+exp_R(-x))/(exp_R
  (x)-exp_R(-x))) by SIN_COS:def 23
    .=(exp_R(x)-exp_R(-x))/(exp_R(x)+exp_R(-x))* ((exp_R(x)+exp_R(-x))/(
  exp_R(x)-exp_R(-x))) by SIN_COS:def 23
    .=(exp_R(x)-exp_R(-x))/(exp_R(x)+exp_R(-x))* (exp_R(x)+exp_R(-x))/(exp_R
  (x)-exp_R(-x)) by XCMPLX_1:74;
  then tanh(x)*coth(x) =(exp_R(x)-exp_R(-x))/(exp_R(x)-exp_R(-x)) by A2,
XCMPLX_1:87
    .=1 by A1,XCMPLX_1:60;
  hence thesis;
end;
