let x be Real; ( 0 < x & x < 1 implies tanh" x = cosh1" (1 / (sqrt (1 - (x ^2)))) )
assume that
A1:
0 < x
and
A2:
x < 1
; tanh" x = cosh1" (1 / (sqrt (1 - (x ^2))))
A3:
1 - (x ^2) > 0
by A1, A2, Lm5;
A4:
(1 + x) / (1 - x) > 0
by A1, A2, Th15;
A5:
1 - x > 0
by A2, XREAL_1:50;
A6:
sqrt (1 + x) > 0
by A1, SQUARE_1:25;
(1 + x) / (1 - x) > 0
by A1, A2, Th15;
then A7:
sqrt ((1 + x) / (1 - x)) = ((1 + x) / (1 - x)) to_power (1 / 2)
by ASYMPT_1:83;
A8:
x ^2 >= 0
by XREAL_1:63;
cosh1" (1 / (sqrt (1 - (x ^2)))) =
log (number_e,((1 / (sqrt (1 - (x ^2)))) + (sqrt ((1 / ((sqrt (1 - (x ^2))) ^2)) - (1 ^2)))))
by XCMPLX_1:76
.=
log (number_e,((1 / (sqrt (1 - (x ^2)))) + (sqrt ((1 / (1 - (x ^2))) - 1))))
by A3, SQUARE_1:def 2
.=
log (number_e,((1 / (sqrt (1 - (x ^2)))) + (sqrt ((1 - (1 * (1 - (x ^2)))) / (1 - (x ^2))))))
by A3, XCMPLX_1:126
.=
log (number_e,((1 / (sqrt (1 - (x ^2)))) + ((sqrt (x ^2)) / (sqrt (1 - (x ^2))))))
by A3, A8, SQUARE_1:30
.=
log (number_e,((1 / (sqrt (1 - (x ^2)))) + (x / (sqrt (1 - (x ^2))))))
by A1, SQUARE_1:22
.=
log (number_e,((1 + x) / (sqrt ((1 - x) * (1 + x)))))
.=
log (number_e,((1 + x) / ((sqrt (1 - x)) * (sqrt (1 + x)))))
by A1, A5, SQUARE_1:29
.=
log (number_e,((sqrt ((1 + x) ^2)) / ((sqrt (1 - x)) * (sqrt (1 + x)))))
by A1, SQUARE_1:22
.=
log (number_e,(((sqrt (1 + x)) * (sqrt (1 + x))) / ((sqrt (1 - x)) * (sqrt (1 + x)))))
by A1, SQUARE_1:29
.=
log (number_e,((sqrt (1 + x)) / (sqrt (1 - x))))
by A6, XCMPLX_1:91
.=
log (number_e,(sqrt ((1 + x) / (1 - x))))
by A1, A5, SQUARE_1:30
.=
(1 / 2) * (log (number_e,((1 + x) / (1 - x))))
by A7, A4, Lm1, POWER:55, TAYLOR_1:11
;
hence
tanh" x = cosh1" (1 / (sqrt (1 - (x ^2))))
; verum