reserve x,y,t for Real;

theorem
  x>1 implies cosh1"(x)=tanh"((sqrt(x^2-1))/x)
proof
  assume
A1: x>1;
  then
A2: sqrt(x^2-1)+x>0 by Th23;
  x^2>1^2+0 by A1,SQUARE_1:16;
  then
A3: x^2-1>0 by XREAL_1:20;
  tanh"((sqrt(x^2-1))/x) =(1/2)*log(number_e,((sqrt(x^2-1)+x*1)/x)/(1-(
  sqrt(x^2-1))/x)) by A1,XCMPLX_1:113
    .=(1/2)*log(number_e,((sqrt(x^2-1)+x)/x)/((1*x-(sqrt(x^2-1)))/x)) by A1,
XCMPLX_1:127
    .=(1/2)*log(number_e,(sqrt(x^2-1)+x)/(x-sqrt(x^2-1))) by A1,XCMPLX_1:55
    .=(1/2)*log(number_e,((sqrt(x^2-1)+x)*(sqrt(x^2-1)+x))/ ((x-sqrt(x^2-1))
  *(sqrt(x^2-1)+x))) by A2,XCMPLX_1:91
    .=(1/2)*log(number_e,((sqrt(x^2-1)+x)*(sqrt(x^2-1)+x))/ (x^2-(sqrt(x^2-1
  ))^2))
    .=(1/2)*log(number_e,((sqrt(x^2-1)+x)*(sqrt(x^2-1)+x))/ (x^2-(x^2-1)))
  by A3,SQUARE_1:def 2
    .=(1/2)*log(number_e,((sqrt(x^2-1)+x)^2))
    .=(1/2)*log(number_e,((sqrt(x^2-1)+x) to_power 2)) by POWER:46
    .=(1/2)*(2*log(number_e,(sqrt(x^2-1)+x))) by A2,Lm1,POWER:55,TAYLOR_1:11
    .=log(number_e,(sqrt(x^2-1)+x));
  hence thesis;
end;
