reserve x,y,t for Real;

theorem
  0<x & x<1 implies tanh"(x)=cosh1"(1/(sqrt(1-x^2)))
proof
  assume that
A1: 0<x and
A2: x<1;
A3: 1-x^2>0 by A1,A2,Lm5;
A4: (1+x)/(1-x)>0 by A1,A2,Th15;
A5: 1-x>0 by A2,XREAL_1:50;
A6: sqrt(1+x)>0 by A1,SQUARE_1:25;
  (1+x)/(1-x)>0 by A1,A2,Th15;
  then
A7: sqrt((1+x)/(1-x))=((1+x)/(1-x)) to_power (1/2) by ASYMPT_1:83;
A8: x^2>=0 by XREAL_1:63;
  cosh1"(1/(sqrt(1-x^2))) =log(number_e,(1/(sqrt(1-x^2)))+sqrt((1/((sqrt(
  1-x^2))^2))-1^2)) by XCMPLX_1:76
    .=log(number_e,(1/(sqrt(1-x^2)))+sqrt(1/(1-x^2)-1)) by A3,SQUARE_1:def 2
    .=log(number_e,(1/(sqrt(1-x^2)))+sqrt((1-1*(1-x^2))/(1-x^2))) by A3,
XCMPLX_1:126
    .=log(number_e,(1/(sqrt(1-x^2)))+sqrt(x^2)/sqrt(1-x^2)) by A3,A8,
SQUARE_1:30
    .=log(number_e,1/(sqrt(1-x^2))+x/sqrt(1-x^2)) by A1,SQUARE_1:22
    .=log(number_e,(1+x)/sqrt((1-x)*(1+x)))
    .=log(number_e,(1+x)/(sqrt(1-x)*sqrt(1+x))) by A1,A5,SQUARE_1:29
    .=log(number_e,(sqrt((1+x)^2))/(sqrt(1-x)*sqrt(1+x))) by A1,SQUARE_1:22
    .=log(number_e,(sqrt(1+x)*sqrt(1+x))/(sqrt(1-x)*sqrt(1+x))) by A1,
SQUARE_1:29
    .=log(number_e,(sqrt(1+x))/(sqrt(1-x))) by A6,XCMPLX_1:91
    .=log(number_e,sqrt((1+x)/(1-x))) by A1,A5,SQUARE_1:30
    .=(1/2)*log(number_e,((1+x)/(1-x))) by A7,A4,Lm1,POWER:55,TAYLOR_1:11;
  hence thesis;
end;
