reserve x,y,t for Real;

theorem
  y=(exp_R(x)+exp_R(-x))/(exp_R(x)-exp_R(-x)) & x<>0 implies x=(1/2)*log
  (number_e,(y+1)/(y-1))
proof
A1: 0<exp_R(x) by SIN_COS:55;
  set t=exp_R x;
  assume that
A2: y=(exp_R(x)+exp_R(-x))/(exp_R(x)-exp_R(-x)) and
A3: x<>0;
  y=(exp_R(x)+1/exp_R(x))/(exp_R(x)-exp_R(-x)) by A2,TAYLOR_1:4;
  then y=(exp_R(x)+1/exp_R(x))/(exp_R(x)-1/exp_R(x)) by TAYLOR_1:4;
  then y=(exp_R(x)+1/exp_R(x))/((exp_R(x)*exp_R(x)-1)/exp_R(x)) by A1,
XCMPLX_1:127;
  then y=((1+exp_R(x)*exp_R(x))/exp_R(x))/(((exp_R(x))^2-1)/exp_R(x)) by A1,
XCMPLX_1:113;
  then
A4: y=(1+(exp_R x)^2)/((exp_R x)^2-1) by A1,XCMPLX_1:55;
  then y*(t^2-1)=1+t^2 by A3,Th30,XCMPLX_1:87;
  then
A5: t^2*(y-1)/(y-1)=(y+1)/(y-1);
  exp_R(x)<>1 by A3,Th29;
  then
A6: 1<y or y< -1 by A4,Lm12,SIN_COS:55;
  then 1-1<y-1 or y-1< -1-1 by XREAL_1:14;
  then sqrt(t^2)=sqrt((y+1)/(y-1)) by A5,XCMPLX_1:89;
  then
A7: exp_R(x)=sqrt((y+1)/(y-1)) by A1,SQUARE_1:22;
A8: 0<(y+1)/(y-1) by A6,Lm7;
  then sqrt((y+1)/(y-1))=((y+1)/(y-1)) to_power (1/2) by ASYMPT_1:83;
  then log(number_e,((y+1)/(y-1)) to_power (1/2))=x by A7,TAYLOR_1:12;
  hence thesis by A8,Lm1,POWER:55,TAYLOR_1:11;
end;
