let x be Real; ( x ^2 < 1 implies tanh" x = sinh" (x / (sqrt (1 - (x ^2)))) )
assume A1:
x ^2 < 1
; tanh" x = sinh" (x / (sqrt (1 - (x ^2))))
then A2:
x + 1 > 0
by Th11;
A3:
sqrt (x + 1) > 0
by A1, Th11, SQUARE_1:25;
A4:
(x + 1) / (1 - x) > 0
by A1, Lm4;
A5:
1 - (x ^2) > 0
by A1, XREAL_1:50;
A6:
1 - x > 0
by A1, Th11;
then A7:
sqrt ((x + 1) / (1 - x)) = ((x + 1) / (1 - x)) to_power (1 / 2)
by A2, ASYMPT_1:83;
sinh" (x / (sqrt (1 - (x ^2)))) =
log (number_e,((x / (sqrt (1 - (x ^2)))) + (sqrt (((x ^2) / ((sqrt (1 - (x ^2))) ^2)) + 1))))
by XCMPLX_1:76
.=
log (number_e,((x / (sqrt (1 - (x ^2)))) + (sqrt (((x ^2) / (1 - (x ^2))) + 1))))
by A5, SQUARE_1:def 2
.=
log (number_e,((x / (sqrt (1 - (x ^2)))) + (sqrt (((x ^2) + ((1 - (x ^2)) * 1)) / (1 - (x ^2))))))
by A5, XCMPLX_1:113
.=
log (number_e,((x / (sqrt (1 - (x ^2)))) + ((sqrt 1) / (sqrt (1 - (x ^2))))))
by A5, SQUARE_1:30
.=
log (number_e,((x + 1) / (sqrt ((1 - x) * (1 + x)))))
by SQUARE_1:18
.=
log (number_e,((sqrt ((x + 1) ^2)) / (sqrt ((1 - x) * (1 + x)))))
by A2, SQUARE_1:22
.=
log (number_e,(((sqrt (x + 1)) * (sqrt (x + 1))) / (sqrt ((1 - x) * (1 + x)))))
by A2, SQUARE_1:29
.=
log (number_e,(((sqrt (x + 1)) * (sqrt (x + 1))) / ((sqrt (1 - x)) * (sqrt (1 + x)))))
by A2, A6, SQUARE_1:29
.=
log (number_e,((sqrt (x + 1)) / (sqrt (1 - x))))
by A3, XCMPLX_1:91
.=
log (number_e,(sqrt ((x + 1) / (1 - x))))
by A2, A6, SQUARE_1:30
.=
(1 / 2) * (log (number_e,((1 + x) / (1 - x))))
by A4, A7, Lm1, POWER:55, TAYLOR_1:11
;
hence
tanh" x = sinh" (x / (sqrt (1 - (x ^2))))
; verum