reserve x,y,t for Real;

theorem
  0<=x implies sinh"(x)=cosh1"(sqrt(x^2+1))
proof
  assume
A1: 0<=x;
  x^2>=0 by XREAL_1:63;
  then cosh1"(sqrt(x^2+1)) =log(number_e,sqrt(x^2+1)+sqrt(x^2+1-1)) by
SQUARE_1:def 2
    .=log(number_e,(sqrt(x^2+1)+x)) by A1,SQUARE_1:22;
  hence thesis;
end;
