reserve x,y,t for Real;

theorem
  y=1/((exp_R(x)-exp_R(-x))/2) & x<>0 implies x=log(number_e,(1+sqrt(1+y
  ^2))/y) or x=log(number_e,(1-sqrt(1+y^2))/y)
proof
A1: 0<exp_R(x) by SIN_COS:55;
  set t=exp_R x;
  assume that
A2: y=1/((exp_R(x)-exp_R(-x))/2) and
A3: x<>0;
A4: delta(y,-2,-y)=(-2)^2-4*y*(-y) by QUIN_1:def 1
    .=4+4*y^2;
  y=(1*2)/(2*((exp_R(x)-exp_R(-x))/2)) by A2,XCMPLX_1:91;
  then y=2/(exp_R(x)-1/exp_R(x)) by TAYLOR_1:4;
  then y=2/((exp_R(x)*exp_R(x)-1)/exp_R(x)) by A1,XCMPLX_1:127;
  then
A5: y=2*(exp_R(x)/((exp_R(x))^2-1)) by XCMPLX_1:79;
  then
A6: y=2*exp_R(x)/((exp_R(x))^2-1);
  then y*(t^2-1)=2*t by A3,Th30,XCMPLX_1:87;
  then
A7: y*t^2+(-2)*t+(-y)=0;
A8: exp_R(x)<>1 by A3,Th29;
  then
A9: 2*exp_R(x)/((exp_R(x))^2-1)<>0 by Lm20,SIN_COS:55;
  per cases by A9,A5;
  suppose
A10: 0<y;
A11: 0<1+y^2 by Lm6;
    then 0*4<4*(1+y^2);
    then
    t=(2+sqrt delta(y,-2,-y))/(2*y) or t=(-(-2)-sqrt delta(y,-2,-y))/(2*y
    ) by A1,A8,A6,A7,A4,Lm20,QUIN_1:15;
    then
    t=(2+sqrt(4)*sqrt(1+y^2))/(2*y) or t=(2-sqrt(4*(1+y^2)))/(2*y) by A4,A11,
SQUARE_1:29;
    then t=(2*(1+sqrt(1+y^2)))/(2*y) or t=(2-2*sqrt(1+y^2))/(2*y) by A11,
SQUARE_1:20,29;
    then
A12: t=(1+sqrt(1+y^2))/y or t=(2*(1-sqrt(1+y^2)))/(2*y) by XCMPLX_1:91;
    (1-sqrt(1+y^2))/y<0 by A10,Lm15;
    hence thesis by A1,A12,TAYLOR_1:12,XCMPLX_1:91;
  end;
  suppose
A13: y<0;
A14: 0<1+y^2 by Lm6;
    then 0*4<4*(1+y^2);
    then
    t=(2+sqrt delta(y,-2,-y))/(2*y) or t=(-(-2)-sqrt delta(y,-2,-y))/(2*y
    ) by A1,A8,A6,A7,A4,Lm20,QUIN_1:15;
    then
    t=(2+sqrt(4)*sqrt(1+y^2))/(2*y) or t=(2-sqrt(4*(1+y^2)))/(2*y) by A4,A14,
SQUARE_1:29;
    then t=(2*(1+sqrt(1+y^2)))/(2*y) or t=(2-2*sqrt(1+y^2))/(2*y) by A14,
SQUARE_1:20,29;
    then
A15: t=(1+sqrt(1+y^2))/y or t=(2*(1-sqrt(1+y^2)))/(2*y) by XCMPLX_1:91;
    (1+sqrt(1+y^2))/y<0 by A13,Lm21;
    then exp_R(x)=(1-sqrt(1+y^2))/y by A15,SIN_COS:55,XCMPLX_1:91;
    hence thesis by TAYLOR_1:12;
  end;
end;
