let x, y be real number ; ( 1 <= x & 1 <= y & abs y <= abs x implies (cosh1" x) - (cosh1" y) = cosh1" ((x * y) - (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1)))) )
assume that
A1:
1 <= x
and
A2:
1 <= y
and
A3:
abs y <= abs x
; (cosh1" x) - (cosh1" y) = cosh1" ((x * y) - (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1))))
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:62, 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_1:92
.=
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 4
.=
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_1:97
.=
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 4
.=
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)))))))
;
(y * (sqrt ((x ^2 ) - 1))) - (x * (sqrt ((y ^2 ) - 1))) =
sqrt (((y * (sqrt ((x ^2 ) - 1))) - (x * (sqrt ((y ^2 ) - 1)))) ^2 )
by A1, A2, A3, Th26, SQUARE_1:89
.=
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 4
.=
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 4
.=
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_1:97
;
hence
(cosh1" x) - (cosh1" y) = cosh1" ((x * y) - (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1))))
by A7, A8; verum