reserve x,y,t for Real;

theorem
  1<=x & 1<=y implies cosh1"(x)+cosh1"(y)=cosh1"(x*y+sqrt((x^2-1)*(y^2-1 )))
proof
  assume that
A1: 1<=x and
A2: 1<=y;
A3: y^2-1>=0 by A2,Lm3;
  set t=x*sqrt(y^2-1)+y*sqrt(x^2-1);
A4: x^2-1>=0 by A1,Lm3;
  t=sqrt((x*sqrt(y^2-1)+y*sqrt(x^2-1))^2) by A1,A2,Th24,SQUARE_1:22
    .=sqrt(x^2*(sqrt(y^2-1))^2+2*(x*sqrt(y^2-1))*(y*sqrt(x^2-1))+ (y*sqrt(x
  ^2-1))^2)
    .=sqrt(x^2*(y^2-1)+2*(x*sqrt(y^2-1))*(y*sqrt(x^2-1))+ (y*sqrt(x^2-1))^2)
  by A3,SQUARE_1:def 2
    .=sqrt(x^2*y^2-x^2+2*(x*sqrt(y^2-1))*(y*sqrt(x^2-1))+ y^2*(sqrt(x^2-1))
  ^2)
    .=sqrt(x^2*y^2-x^2+2*(x*sqrt(y^2-1))*(y*sqrt(x^2-1))+ y^2*(x^2-1)) by A4,
SQUARE_1:def 2
    .=sqrt(2*(x*y)^2-x^2-y^2+2*(x*sqrt(y^2-1))*(y*sqrt(x^2-1)));
  then
A5: log(number_e,x*sqrt(y^2-1)+y*sqrt(x^2-1)+sqrt((x^2-1)*(y^2-1))+x*y) =log
(number_e,(sqrt(2*(x*y)^2-x^2-y^2+2*x*y*(sqrt(y^2-1) *sqrt(x^2-1))))+sqrt((x^2-
  1)*(y^2-1))+x*y)
    .=log(number_e,(sqrt(2*(x*y)^2-x^2-y^2+2*x*y*sqrt((y^2-1)*(x^2-1)))) +
  sqrt((x^2-1)*(y^2-1))+x*y) by A4,A3,SQUARE_1:29;
A6: cosh1"(x*y+sqrt((x^2-1)*(y^2-1))) =log(number_e,(x*y+sqrt((x^2-1)*(y^2-
1)))+sqrt((x*y)^2+ 2*(x*y)*(sqrt((x^2-1)*(y^2-1)))+((sqrt((x^2-1)*(y^2-1)))^2)-
  1))
    .=log(number_e,(x*y+sqrt((x^2-1)*(y^2-1)))+sqrt((x*y)^2+ 2*(x*y)*(sqrt((
  x^2-1)*(y^2-1)))+(x^2-1)*(y^2-1)-1)) by A4,A3,SQUARE_1:def 2
    .=log(number_e,((sqrt(2*(x*y)^2-x^2-y^2+2*x*y*sqrt((y^2-1)*(x^2-1))))) +
  (sqrt((x^2-1)*(y^2-1))+x*y));
  0<x+sqrt(x^2-1) & 0<y+sqrt(y^2-1) by A1,A2,Th23;
  then cosh1"(x)+cosh1"(y) =log(number_e,(x+sqrt(x^2-1))*(y+sqrt(y^2-1))) by
Lm1,POWER:53,TAYLOR_1:11
    .=log(number_e,x*sqrt(y^2-1)+y*sqrt(x^2-1)+(sqrt(x^2-1))*(sqrt(y^2-1))+x
  *y)
    .=log(number_e,x*sqrt(y^2-1)+y*sqrt(x^2-1)+sqrt((x^2-1)*(y^2-1))+x*y) by A4
,A3,SQUARE_1:29;
  hence thesis by A5,A6;
end;
