reserve x,y,t for Real;

theorem
  1<=x & 1<=y & |.y.|<=|.x.| 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 and
A3: |.y.|<=|.x.|;
A4: 0<=x^2-1 by A1,Lm3;
  set t=y*sqrt(x^2-1)-x*sqrt(y^2-1);
A5: (y-sqrt(y^2-1))>0 by A2,Th25;
A6: 0<=y^2-1 by A2,Lm3;
  0<x+sqrt(x^2-1) & 0<y+sqrt(y^2-1) by A1,A2,Th23;
  then
A7: cosh1"(x)-cosh1"(y) =log(number_e,((x+sqrt(x^2-1))/(y+sqrt(y^2-1)))) by Lm1
,POWER:54,TAYLOR_1:11
    .=log(number_e,(((x+sqrt(x^2-1))*(y-sqrt(y^2-1)))/ ((y+sqrt(y^2-1))*(y-
  sqrt(y^2-1))))) by A5,XCMPLX_1:91
    .=log(number_e,(((x+sqrt(x^2-1))*(y-sqrt(y^2-1)))/ (y^2-(sqrt(y^2-1))^2)
  ))
    .=log(number_e,(((x+sqrt(x^2-1))*(y-sqrt(y^2-1)))/ (y^2-(y^2-1)))) by A6,
SQUARE_1:def 2
    .=log(number_e,(x*y-x*sqrt(y^2-1)+y*sqrt(x^2-1)-sqrt(x^2-1)* (sqrt(y^2-1
  ))))
    .=log(number_e,(x*y-x*sqrt(y^2-1)+y*sqrt(x^2-1)-sqrt((x^2-1)*(y^2-1))))
  by A4,A6,SQUARE_1:29
    .=log(number_e,(x*y-sqrt((x^2-1)*(y^2-1))+y*sqrt(x^2-1)-x*sqrt(y^2-1)));
A8: 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,A6,SQUARE_1:def 2
    .=log(number_e,(x*y-sqrt((x^2-1)*(y^2-1)))+sqrt(2*(x*y)^2-x^2-y^2 -2*x*y
  *(sqrt((x^2-1)*(y^2-1)))));
  t=sqrt((y*sqrt(x^2-1)-x*sqrt(y^2-1))^2) by A1,A2,A3,Th26,SQUARE_1:22
    .=sqrt(y^2*(sqrt(x^2-1))^2-2*(y*sqrt(x^2-1))* (x*sqrt(y^2-1))+(x*sqrt(y
  ^2-1))^2)
    .=sqrt(y^2*(x^2-1)-2*(y*sqrt(x^2-1))* (x*sqrt(y^2-1))+(x*sqrt(y^2-1))^2)
  by A4,SQUARE_1:def 2
    .=sqrt((x*y)^2-y^2-2*(y*sqrt(x^2-1))* (x*sqrt(y^2-1))+x^2*(sqrt(y^2-1))
  ^2)
    .=sqrt((x*y)^2-y^2-2*(y*sqrt(x^2-1))* (x*sqrt(y^2-1))+x^2*(y^2-1)) by A6,
SQUARE_1:def 2
    .=sqrt(2*(x*y)^2-x^2-y^2-2*x*y*(sqrt(x^2-1)*sqrt(y^2-1)))
    .=sqrt(2*(x*y)^2-x^2-y^2-2*x*y*(sqrt((x^2-1)*(y^2-1)))) by A4,A6,
SQUARE_1:29;
  hence thesis by A7,A8;
end;
