reserve x,y,t for Real;

theorem
  y=1/((exp_R(x)+exp_R(-x))/2) implies x=log(number_e,(1+sqrt(1-y^2))/y)
  or x=-log(number_e,(1+sqrt(1-y^2))/y)
proof
  set t=exp_R(x);
A1: delta(y,-2,y) =(-2)^2-4*y*y by QUIN_1:def 1
    .=4-4*y^2;
  assume y=1/((exp_R(x)+exp_R(-x))/2);
  then y=(1*2)/(2*((exp_R(x)+exp_R(-x))/2)) by XCMPLX_1:91;
  then 0<exp_R(x) & y=2/(exp_R(x)+1/exp_R(x)) by SIN_COS:55,TAYLOR_1:4;
  then y=2/((1+exp_R(x)*exp_R(x))/exp_R(x)) by XCMPLX_1:113;
  then y=2*(exp_R(x)/(1+(exp_R(x))^2)) by XCMPLX_1:79;
  then
A2: y=2*t/(1+t^2);
  then
A3: 0<y by Lm13,SIN_COS:55;
  1+t^2>0 by Lm6;
  then
A4: y*(1+t^2)=(2*t) by A2,XCMPLX_1:87;
A5: y<=1 by A2,Lm14,SIN_COS:55;
  then
A6: 0<=1-y^2 by A3,Lm16;
  Polynom(y,-2,y,t)=y*t^2+(-2)*t+y by POLYEQ_1:def 2;
  then
  t=(-(-2)+sqrt delta(y,-2,y))/(2*y) or t=(-(-2)-sqrt delta(y,-2,y))/(2*y
  ) by A3,A5,A4,A1,Lm17,QUIN_1:15;
  then t=(2+sqrt(4*(1-y^2)))/(2*y) or t=(2-sqrt(4*(1-y^2)))/(2*y) by A1;
  then t=(2+2*sqrt(1-y^2))/(2*y) or t=(2-2*sqrt(1-y^2))/(2*y) by A6,SQUARE_1:20
,29;
  then t=(2*(1+sqrt(1-y^2)))/(2*y) or t=(2*(1-sqrt(1-y^2)))/(2*y);
  then
A7: t=(1+sqrt(1-y^2))/y or t=(1-sqrt(1-y^2))/y by XCMPLX_1:91;
  0<1+sqrt(1-y^2) by A3,A5,Lm18;
  then
  t=(1+sqrt(1-y^2))/y or t=((1-sqrt(1-y^2))*(1+sqrt(1-y^2)))/(y*(1+sqrt(1
  -y^2))) by A7,XCMPLX_1:91;
  then t=(1+sqrt(1-y^2))/y or t=(1-(sqrt(1-y^2))^2)/(y*(1+sqrt(1-y^2)));
  then t=(1+sqrt(1-y^2))/y or t=(1-(1-y^2))/(y*(1+sqrt(1-y^2))) by A6,
SQUARE_1:def 2;
  then
A8: t=(1+sqrt(1-y^2))/y or t=y/(1+sqrt(1-y^2)) by A3,XCMPLX_1:91;
  log(number_e,exp_R(x))=x & 1/((1+sqrt(1-y^2))/y)=((1+sqrt(1-y^2))/y)
  to_power (-1) by A3,A5,Lm19,Th1,TAYLOR_1:12;
  then
A9: log(number_e,(1+sqrt(1-y^2))/y)=x or log(number_e,((1+sqrt(1-y^2))/y)
  to_power (-1))=x by A8,XCMPLX_1:57;
  0<(1+sqrt(1-y^2))/y by A3,A5,Lm19;
  then
  log(number_e,(1+sqrt(1-y^2))/y)=x or (-1)*log(number_e,((1+sqrt(1-y^2))
  /y))=x by A9,Lm1,POWER:55,TAYLOR_1:11;
  hence thesis;
end;
