reserve x,y,t for Real;

theorem
  x^2<1 implies tanh"(x)=(1/2)*sinh"((2*x)/(1-x^2))
proof
  assume
A1: x^2<1;
  then
A2: (1-x^2)^2>0 by Th12;
A3: x+1>0 by A1,Th11;
  (1/2)*sinh"((2*x)/(1-x^2)) =(1/2)*log(number_e,(((2*x)/(1-x^2))+sqrt(((2
  *x)^2/(1-x^2)^2)+1))) by XCMPLX_1:76
    .=(1/2)*log(number_e,(((2*x)/(1-x^2))+sqrt(((4*x^2)+((1-x^2)^2)*1)/ ((1-
  x^2)^2)))) by A2,XCMPLX_1:113
    .=(1/2)*log(number_e,(((2*x)/(1-x^2))+sqrt((((x^2)+1)^2)/ ((1-x^2)^2))))
    .=(1/2)*log(number_e,(((2*x)/(1-x^2))+sqrt((((x^2)+1)/(1-x^2))^2))) by
XCMPLX_1:76
    .=(1/2)*log(number_e,((2*x)/(1-x^2)+((x^2)+1)/(1-x^2))) by A1,Th13,
SQUARE_1:22
    .=(1/2)*log(number_e,(((2*x)+((x^2)+1))/(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 A3,XCMPLX_1:91;
  hence thesis;
end;
