reserve x,y,t for Real;

theorem
  y=(1/2)*(exp_R(x)+exp_R(-x)) & 1<=y implies x=log(number_e,(y+sqrt(y^2
  -1))) or x=-log(number_e,(y+sqrt(y^2-1)))
proof
  assume that
A1: y=(1/2)*(exp_R(x)+exp_R(-x)) and
A2: 1<=y;
A3: y+sqrt(y^2-1)>0 by A2,Lm10;
  set t=exp_R(x);
  2*y*exp_R(x)=(exp_R(x)+1/exp_R(x))*exp_R(x) by A1,TAYLOR_1:4;
  then 0<exp_R(x) & 2*y*t=t^2+(t*1)/t by SIN_COS:55;
  then 2*y*t-2*y*t=t^2+1-2*y*t by XCMPLX_1:60;
  then
A4: 0=1*t^2+(-2*y)*t+1;
A5: delta(1,-2*y,1)=((-2)*y)^2-4*1*1 by QUIN_1:def 1
    .=4*(y^2)-4;
A6: 0<=(y^2)-1 by A2,Lm3;
  then 0*4<=((y^2)-1)*4;
  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 A4,A5,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 A6,A5,
SQUARE_1:29;
  then t=(2*y+2*sqrt(y^2-1))/2 or t=(2*y-2*sqrt(y^2-1))/2 by A6,SQUARE_1:20,29;
  then log(number_e,y+sqrt(y^2-1))=x or log(number_e,y-sqrt(y^2-1))=x by
TAYLOR_1:12;
  then
  log(number_e,y+sqrt(y^2-1))=x or log(number_e,(y-sqrt(y^2-1))*(y+sqrt(y
  ^2-1))/ (y+sqrt(y^2-1)))=x by A3,XCMPLX_1:89;
  then log(number_e,y+sqrt(y^2-1)) = x or log(number_e,(y^2-(sqrt(y^2-1))^2)/
  (y+sqrt(y^2-1)))=x;
  then
A7: log(number_e,y+sqrt(y^2-1))=x or log(number_e,(y^2-(y^2-1))/(y+sqrt(y^2
  -1)))=x by A6,SQUARE_1:def 2;
  1/(y+sqrt(y^2-1))=(y+sqrt(y^2-1)) to_power (-1) by A3,Th1;
  then log(number_e,y+sqrt(y^2-1))=x or (-1)*log(number_e,(y+sqrt(y^2-1)))=x
  by A3,A7,Lm1,POWER:55,TAYLOR_1:11;
  hence thesis;
end;
