reserve x,y,t for Real;

theorem
  x>0 & x<1 implies tanh"(x)=(1/2)*cosh1"((1+x^2)/(1-x^2))
proof
  assume that
A1: x>0 and
A2: x<1;
A3: 0<(1-x^2)^2 by A1,A2,Th19;
A4: (2*x)/(1-x^2)>0 by A1,A2,Th18;
  (1/2)*cosh1"((1+x^2)/(1-x^2)) =(1/2)*log(number_e,((1+x^2)/(1-x^2))+sqrt
  ((1+x^2)^2/(1-x^2)^2-1)) by XCMPLX_1:76
    .=(1/2)*log(number_e,((1+x^2)/(1-x^2))+sqrt((((1+x^2)^2)-1* ((1-x^2)^2))
  /((1-x^2)^2))) by A3,XCMPLX_1:126
    .=(1/2)*log(number_e,((1+x^2)/(1-x^2))+sqrt(((2*x)^2)/((1-x^2)^2)))
    .=(1/2)*log(number_e,((1+x^2)/(1-x^2))+sqrt(((2*x)/(1-x^2))^2)) by
XCMPLX_1:76
    .=(1/2)*log(number_e,(1+x^2)/(1-x^2)+(2*x)/(1-x^2)) by A4,SQUARE_1:22
    .=(1/2)*log(number_e,((1+x^2)+(2*x))/(1-x^2))
    .=(1/2)*log(number_e,(x+1)*(x+1)/((1-x)*(1+x)))
    .=(1/2)*log(number_e,(x+1)/((1-x))) by A1,XCMPLX_1:91;
  hence thesis;
end;
