reserve x,y,t for Real;

theorem
  y=(exp_R(x)-exp_R(-x))/(exp_R(x)+exp_R(-x)) implies x=(1/2)*log(
  number_e,(1+y)/(1-y))
proof
A1: 0<exp_R(x) by SIN_COS:55;
  set t=exp_R(x);
  assume y=(exp_R(x)-exp_R(-x))/(exp_R(x)+exp_R(-x));
  then y=(exp_R(x)-1/exp_R(x))/(exp_R(x)+exp_R(-x)) by 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)*exp_R(x)-1)/exp_R(x))/(exp_R(x)+1/exp_R(x)) by A1,
XCMPLX_1:127;
  then y=(((exp_R x)^2-1)/exp_R(x))/((1+exp_R(x)*exp_R(x))/exp_R(x)) by A1,
XCMPLX_1:113;
  then
A2: y=((exp_R x)^2-1)/(1+(exp_R(x))^2) by A1,XCMPLX_1:55;
  then 1*y+t^2*y=(t^2-1)/(1+t^2)*(1+t^2);
  then y+t^2*y-(t^2-1)=t^2-1-(t^2-1) by A1,XCMPLX_1:87;
  then t^2*(y-1)/(y-1)=(-(y+1))/(y-1);
  then
A3: t^2*(y-1)/(y-1)=(y+1)/(-(y-1)) by XCMPLX_1:192;
  y-1<>0 by A2,Th31;
  then sqrt(t^2)=sqrt((y+1)/(1-y)) by A3,XCMPLX_1:89;
  then
A4: exp_R(x)=sqrt((y+1)/(1-y)) by A1,SQUARE_1:22;
  -1<y by A2,Lm11,SIN_COS:55;
  then
A5: 0<(y+1)/(1-y) by A2,Th31,Th32;
  then sqrt((y+1)/(1-y))=((y+1)/(1-y)) to_power (1/2) by ASYMPT_1:83;
  then log(number_e,((y+1)/(1-y)) to_power (1/2))=x by A4,TAYLOR_1:12;
  hence thesis by A5,Lm1,POWER:55,TAYLOR_1:11;
end;
