reserve x,y,t for Real;

theorem
  x^2<1 implies tanh"(x)=(1/2)*tanh"((2*x)/(1+x^2))
proof
  assume x^2<1;
  then
A1: (x+1)/(1-x)>0 by Lm4;
A2: 1+x^2>0 by Lm6;
  then
  (1/2)*tanh"((2*x)/(1+x^2)) =(1/2)*((1/2)*log(number_e,(((2*x)+(1+x^2)*1)
  /(1+x^2))/(1-(2*x)/(1+x^2)))) by XCMPLX_1:113
    .=(1/2)*((1/2)*log(number_e,((2*x+1+x^2)/(1+x^2))/((1*(1+x^2)-(2*x)) /(1
  +x^2)))) by A2,XCMPLX_1:127
    .=(1/2)*((1/2)*log(number_e,(x+1)^2/(1-x)^2)) by A2,XCMPLX_1:55
    .=(1/2)*((1/2)*log(number_e,((x+1)/(1-x))^2)) by XCMPLX_1:76
    .=(1/2)*((1/2)*log(number_e,((x+1)/(1-x)) to_power 2)) by POWER:46
    .=(1/2)*((1/2)*(2*log(number_e,((x+1)/(1-x))))) by A1,Lm1,POWER:55
,TAYLOR_1:11
    .=(1/2)*log(number_e,(1+x)/(1-x));
  hence thesis;
end;
