reserve x,y,t for Real;

theorem
  x<0 implies sinh"(x)=cosh2"(sqrt(x^2+1))
proof
  assume
A1: x<0;
A2: sqrt(x^2+1)+x>0 by Th5;
A3: 1/(sqrt(x^2+1)+x)=(sqrt(x^2+1)+x) to_power (-1) by Th1,Th5;
A4: x^2>=0 by XREAL_1:63;
  then cosh2"(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_1:23
    .=-log(number_e,(sqrt(x^2+1)-x)*(sqrt(x^2+1)+x)/(sqrt(x^2+1)+x)) by A2,
XCMPLX_1:89
    .=-log(number_e,((sqrt(x^2+1))^2-x^2)/(sqrt(x^2+1)+x))
    .=-log(number_e,((x^2+1)-x^2)/(sqrt(x^2+1)+x)) by A4,SQUARE_1:def 2
    .=-((-1)*log(number_e,(sqrt(x^2+1)+x))) by A2,A3,Lm1,POWER:55,TAYLOR_1:11
    .=log(number_e,(sqrt(x^2+1)+x));
  hence thesis;
end;
