let x be real number ; ( x ^2 < 1 implies tanh" x = (1 / 2) * (sinh" ((2 * x) / (1 - (x ^2 )))) )
assume A1:
x ^2 < 1
; tanh" x = (1 / 2) * (sinh" ((2 * x) / (1 - (x ^2 ))))
then A2:
(1 - (x ^2 )) ^2 > 0
by Th12;
A3:
x + 1 > 0
by A1, Th11;
(1 / 2) * (sinh" ((2 * x) / (1 - (x ^2 )))) =
(1 / 2) * (log number_e ,(((2 * x) / (1 - (x ^2 ))) + (sqrt ((((2 * x) ^2 ) / ((1 - (x ^2 )) ^2 )) + 1))))
by XCMPLX_1:77
.=
(1 / 2) * (log number_e ,(((2 * x) / (1 - (x ^2 ))) + (sqrt (((4 * (x ^2 )) + (((1 - (x ^2 )) ^2 ) * 1)) / ((1 - (x ^2 )) ^2 )))))
by A2, XCMPLX_1:114
.=
(1 / 2) * (log number_e ,(((2 * x) / (1 - (x ^2 ))) + (sqrt ((((x ^2 ) + 1) ^2 ) / ((1 - (x ^2 )) ^2 )))))
.=
(1 / 2) * (log number_e ,(((2 * x) / (1 - (x ^2 ))) + (sqrt ((((x ^2 ) + 1) / (1 - (x ^2 ))) ^2 ))))
by XCMPLX_1:77
.=
(1 / 2) * (log number_e ,(((2 * x) / (1 - (x ^2 ))) + (((x ^2 ) + 1) / (1 - (x ^2 )))))
by A1, Th13, SQUARE_1:89
.=
(1 / 2) * (log number_e ,(((2 * x) + ((x ^2 ) + 1)) / (1 - (x ^2 ))))
.=
(1 / 2) * (log number_e ,(((x + 1) * (x + 1)) / ((1 - x) * (1 + x))))
.=
(1 / 2) * (log number_e ,((x + 1) / (1 - x)))
by A3, XCMPLX_1:92
;
hence
tanh" x = (1 / 2) * (sinh" ((2 * x) / (1 - (x ^2 ))))
; verum