reserve x,y,t for Real;

theorem
  y=(1/2)*(exp_R(x)-exp_R(-x)) implies x=log(number_e,(y+sqrt(y^2+1)))
proof
A1: exp_R(x)>0 by SIN_COS:55;
  set t=exp_R(x);
A2: delta(1,-2*y,-1)=((-2)*y)^2-4*1*(-1) by QUIN_1:def 1
    .=4*(y^2)+4;
A3: 0<=y^2 by XREAL_1:63;
  assume y=(1/2)*(exp_R(x)-exp_R(-x));
  then 2*y*exp_R(x)=(exp_R(x)-1/exp_R(x))*exp_R(x) by TAYLOR_1:4;
  then 2*y*t=t^2-(t*1)/t;
  then 2*y*t-2*y*t=t^2-1-2*y*t by A1,XCMPLX_1:60;
  then 1*t^2+(-2*y)*t+(-1)=0;
  then
  t = (-(-2*y)+sqrt delta(1,-2*y,-1))/(2*1) or t = (-(-2*y)-sqrt delta(1,-
  2*y,-1))/(2*1) by A2,A3,QUIN_1:15;
  then t=(2*y+sqrt(4)*sqrt(y^2+1))/2 or t=(2*y-sqrt(4*(y^2+1)))/2 by A2,A3,
SQUARE_1:29;
  then t=(2*y+2*sqrt(y^2+1))/2 or t=(2*y-2*sqrt(y^2+1))/2 by A3,SQUARE_1:20,29;
  then
A4: exp_R(x)=y+sqrt(y^2+1) or exp_R(x)=y-sqrt(y^2+1);
  y<sqrt(y^2+1)+0 by Lm8;
  hence thesis by A1,A4,TAYLOR_1:12,XREAL_1:19;
end;
