reserve x,y,t for Real;

theorem
  0<x implies log(number_e,x)=tanh"((x^2-1)/(x^2+1))
proof
  assume
A1: 0<x;
A2: x^2+1>0 by Lm6;
  then
  tanh"((x^2-1)/(x^2+1)) =(1/2)*log(number_e,(((x^2-1)+(x^2+1)*1)/(x^2+1))
  /(1-(x^2-1)/(x^2+1))) by XCMPLX_1:113
    .=(1/2)*log(number_e,((2*x^2)/(x^2+1))/((1*(x^2+1)-(x^2-1))/(x^2+1))) by A2
,XCMPLX_1:127
    .=(1/2)*log(number_e,2*x^2/2) by A2,XCMPLX_1:55
    .=(1/2)*log(number_e,x to_power 2) by POWER:46
    .=(1/2)*(2*log(number_e,x)) by A1,Lm1,POWER:55,TAYLOR_1:11
    .=log(number_e,x);
  hence thesis;
end;
