reserve x,y,t for Real;

theorem
  sinh"(x)=tanh"(x/sqrt(x^2+1))
proof
  set t = sqrt(x^2+1)+x;
A1: sqrt(x^2+1)+x>0 by Th5;
A2: x^2+1>0 by Lm6;
A3: sqrt(x^2+1)>0 by Th4;
  then
  tanh"(x/sqrt(x^2+1)) =(1/2)*log(number_e,((sqrt(x^2+1)*1+x)/sqrt(x^2+1))
  /(1-x/sqrt(x^2+1))) by XCMPLX_1:113
    .=(1/2)*log(number_e,((sqrt(x^2+1)*1+x)/sqrt(x^2+1))/((1*sqrt(x^2+1)-x)
  /sqrt(x^2+1))) by A3,XCMPLX_1:127
    .=(1/2)*log(number_e,t/(sqrt(x^2+1)-x)) by A3,XCMPLX_1:55
    .=(1/2)*log(number_e,t*t/((sqrt(x^2+1)-x)*t)) by A1,XCMPLX_1:91
    .=(1/2)*log(number_e,(t*t)/((sqrt(x^2+1))*(sqrt(x^2+1))-x^2))
    .=(1/2)*log(number_e,(t*t)/((sqrt((x^2+1)^2))-x^2)) by A2,SQUARE_1:29
    .=(1/2)*log(number_e,(t*t)/((x^2+1)-x^2)) by A2,SQUARE_1:22
    .=(1/2)*log(number_e,t^2)
    .=(1/2)*log(number_e,t to_power 2) by POWER:46
    .=(1/2)*(2*log(number_e,t)) by A1,Lm1,POWER:55,TAYLOR_1:11
    .=log(number_e,t);
  hence thesis;
end;
