reserve x,y,t for Real;

theorem
  x>=1 implies cosh1"(x)=2*sinh"(sqrt((x-1)/2))
proof
  assume
A1: x>=1;
  then
A2: sqrt((x+1)/2)+sqrt((x-1)/2)>0 by Th10;
A3: (x^2-1)/4>=0 by A1,Th9;
A4: (x-1)/2>=0 by A1,Th7;
  then
  2*sinh"(sqrt((x-1)/2)) =2*log(number_e,((sqrt((x-1)/2))+sqrt(((x-1)/2)+1
  ))) by SQUARE_1:def 2
    .=log(number_e,(sqrt((x-1)/2)+sqrt((x+1)/2)) to_power 2) by A2,Lm1,POWER:55
,TAYLOR_1:11
    .=log(number_e,(sqrt((x-1)/2)+sqrt((x+1)/2))^2) by POWER:46
    .=log(number_e,(sqrt((x-1)/2))^2+2*(sqrt((x-1)/2)) *(sqrt((x+1)/2))+(
  sqrt((x+1)/2))^2)
    .=log(number_e,((x-1)/2)+2*(sqrt((x-1)/2)) *(sqrt((x+1)/2))+(sqrt((x+1)/
  2))^2) by A4,SQUARE_1:def 2
    .=log(number_e,(x-1)/2+2*(sqrt((x-1)/2)) *(sqrt((x+1)/2))+(x+1)/2) by A1,
SQUARE_1:def 2
    .=log(number_e,x+2*((sqrt((x-1)/2))*(sqrt((x+1)/2))))
    .=log(number_e,x+2*(sqrt(((x-1)/2)*((x+1)/2)))) by A1,A4,SQUARE_1:29
    .=log(number_e,x+sqrt(2^2)*(sqrt((x^2-1)/4))) by SQUARE_1:22
    .=log(number_e,x+sqrt(4*((x^2-1)/4))) by A3,SQUARE_1:29
    .=log(number_e,x+sqrt(x^2-1));
  hence thesis;
end;
