let r be Element of REAL ; for f1, f2, f3, g1, g2, g3 being PartFunc of REAL ,REAL
for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 holds
VFuncdiff ((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0 = r * |[((((diff f1,t0) * (g1 . t0)) - ((diff g1,t0) * (f1 . t0))) / ((g1 . t0) ^2 )),((((diff f2,t0) * (g2 . t0)) - ((diff g2,t0) * (f2 . t0))) / ((g2 . t0) ^2 )),((((diff f3,t0) * (g3 . t0)) - ((diff g3,t0) * (f3 . t0))) / ((g3 . t0) ^2 ))]|
let f1, f2, f3, g1, g2, g3 be PartFunc of REAL ,REAL ; for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 holds
VFuncdiff ((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0 = r * |[((((diff f1,t0) * (g1 . t0)) - ((diff g1,t0) * (f1 . t0))) / ((g1 . t0) ^2 )),((((diff f2,t0) * (g2 . t0)) - ((diff g2,t0) * (f2 . t0))) / ((g2 . t0) ^2 )),((((diff f3,t0) * (g3 . t0)) - ((diff g3,t0) * (f3 . t0))) / ((g3 . t0) ^2 ))]|
let t0 be Real; ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 & g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 implies VFuncdiff ((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0 = r * |[((((diff f1,t0) * (g1 . t0)) - ((diff g1,t0) * (f1 . t0))) / ((g1 . t0) ^2 )),((((diff f2,t0) * (g2 . t0)) - ((diff g2,t0) * (f2 . t0))) / ((g2 . t0) ^2 )),((((diff f3,t0) * (g3 . t0)) - ((diff g3,t0) * (f3 . t0))) / ((g3 . t0) ^2 ))]| )
assume that
A1:
( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 )
and
A2:
( g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 )
and
A3:
( g1 . t0 <> 0 & g2 . t0 <> 0 & g3 . t0 <> 0 )
; VFuncdiff ((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0 = r * |[((((diff f1,t0) * (g1 . t0)) - ((diff g1,t0) * (f1 . t0))) / ((g1 . t0) ^2 )),((((diff f2,t0) * (g2 . t0)) - ((diff g2,t0) * (f2 . t0))) / ((g2 . t0) ^2 )),((((diff f3,t0) * (g3 . t0)) - ((diff g3,t0) * (f3 . t0))) / ((g3 . t0) ^2 ))]|
A4:
( r (#) f1 is_differentiable_in t0 & r (#) f2 is_differentiable_in t0 & r (#) f3 is_differentiable_in t0 )
by A1, FDIFF_1:23;
then VFuncdiff ((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0 =
|[((((diff (r (#) f1),t0) * (g1 . t0)) - ((diff g1,t0) * ((r (#) f1) . t0))) / ((g1 . t0) ^2 )),(diff ((r (#) f2) / g2),t0),(diff ((r (#) f3) / g3),t0)]|
by A2, A3, FDIFF_2:14
.=
|[((((diff (r (#) f1),t0) * (g1 . t0)) - ((diff g1,t0) * ((r (#) f1) . t0))) / ((g1 . t0) ^2 )),((((diff (r (#) f2),t0) * (g2 . t0)) - ((diff g2,t0) * ((r (#) f2) . t0))) / ((g2 . t0) ^2 )),(diff ((r (#) f3) / g3),t0)]|
by A2, A3, A4, FDIFF_2:14
.=
|[((((diff (r (#) f1),t0) * (g1 . t0)) - ((diff g1,t0) * ((r (#) f1) . t0))) / ((g1 . t0) ^2 )),((((diff (r (#) f2),t0) * (g2 . t0)) - ((diff g2,t0) * ((r (#) f2) . t0))) / ((g2 . t0) ^2 )),((((diff (r (#) f3),t0) * (g3 . t0)) - ((diff g3,t0) * ((r (#) f3) . t0))) / ((g3 . t0) ^2 ))]|
by A2, A3, A4, FDIFF_2:14
.=
|[((((r * (diff f1,t0)) * (g1 . t0)) - ((diff g1,t0) * ((r (#) f1) . t0))) / ((g1 . t0) ^2 )),((((diff (r (#) f2),t0) * (g2 . t0)) - ((diff g2,t0) * ((r (#) f2) . t0))) / ((g2 . t0) ^2 )),((((diff (r (#) f3),t0) * (g3 . t0)) - ((diff g3,t0) * ((r (#) f3) . t0))) / ((g3 . t0) ^2 ))]|
by A1, FDIFF_1:23
.=
|[((((r * (diff f1,t0)) * (g1 . t0)) - ((diff g1,t0) * ((r (#) f1) . t0))) / ((g1 . t0) ^2 )),((((r * (diff f2,t0)) * (g2 . t0)) - ((diff g2,t0) * ((r (#) f2) . t0))) / ((g2 . t0) ^2 )),((((diff (r (#) f3),t0) * (g3 . t0)) - ((diff g3,t0) * ((r (#) f3) . t0))) / ((g3 . t0) ^2 ))]|
by A1, FDIFF_1:23
.=
|[((((r * (diff f1,t0)) * (g1 . t0)) - ((diff g1,t0) * ((r (#) f1) . t0))) / ((g1 . t0) ^2 )),((((r * (diff f2,t0)) * (g2 . t0)) - ((diff g2,t0) * ((r (#) f2) . t0))) / ((g2 . t0) ^2 )),((((r * (diff f3,t0)) * (g3 . t0)) - ((diff g3,t0) * ((r (#) f3) . t0))) / ((g3 . t0) ^2 ))]|
by A1, FDIFF_1:23
.=
|[((((r * (diff f1,t0)) * (g1 . t0)) - ((diff g1,t0) * (r * (f1 . t0)))) / ((g1 . t0) ^2 )),((((r * (diff f2,t0)) * (g2 . t0)) - ((diff g2,t0) * ((r (#) f2) . t0))) / ((g2 . t0) ^2 )),((((r * (diff f3,t0)) * (g3 . t0)) - ((diff g3,t0) * ((r (#) f3) . t0))) / ((g3 . t0) ^2 ))]|
by VALUED_1:6
.=
|[((((r * (diff f1,t0)) * (g1 . t0)) - ((diff g1,t0) * (r * (f1 . t0)))) / ((g1 . t0) ^2 )),((((r * (diff f2,t0)) * (g2 . t0)) - ((diff g2,t0) * (r * (f2 . t0)))) / ((g2 . t0) ^2 )),((((r * (diff f3,t0)) * (g3 . t0)) - ((diff g3,t0) * ((r (#) f3) . t0))) / ((g3 . t0) ^2 ))]|
by VALUED_1:6
.=
|[((((r * (diff f1,t0)) * (g1 . t0)) - ((diff g1,t0) * (r * (f1 . t0)))) / ((g1 . t0) ^2 )),((((r * (diff f2,t0)) * (g2 . t0)) - ((diff g2,t0) * (r * (f2 . t0)))) / ((g2 . t0) ^2 )),((((r * (diff f3,t0)) * (g3 . t0)) - ((diff g3,t0) * (r * (f3 . t0)))) / ((g3 . t0) ^2 ))]|
by VALUED_1:6
.=
|[((r * (((diff f1,t0) * (g1 . t0)) - ((diff g1,t0) * (f1 . t0)))) / ((g1 . t0) ^2 )),((r * (((diff f2,t0) * (g2 . t0)) - ((diff g2,t0) * (f2 . t0)))) / ((g2 . t0) ^2 )),((r * (((diff f3,t0) * (g3 . t0)) - ((diff g3,t0) * (f3 . t0)))) / ((g3 . t0) ^2 ))]|
.=
|[(r * ((((diff f1,t0) * (g1 . t0)) - ((diff g1,t0) * (f1 . t0))) / ((g1 . t0) ^2 ))),((r * (((diff f2,t0) * (g2 . t0)) - ((diff g2,t0) * (f2 . t0)))) / ((g2 . t0) ^2 )),((r * (((diff f3,t0) * (g3 . t0)) - ((diff g3,t0) * (f3 . t0)))) / ((g3 . t0) ^2 ))]|
by XCMPLX_1:75
.=
|[(r * ((((diff f1,t0) * (g1 . t0)) - ((diff g1,t0) * (f1 . t0))) / ((g1 . t0) ^2 ))),(r * ((((diff f2,t0) * (g2 . t0)) - ((diff g2,t0) * (f2 . t0))) / ((g2 . t0) ^2 ))),((r * (((diff f3,t0) * (g3 . t0)) - ((diff g3,t0) * (f3 . t0)))) / ((g3 . t0) ^2 ))]|
by XCMPLX_1:75
.=
|[(r * ((((diff f1,t0) * (g1 . t0)) - ((diff g1,t0) * (f1 . t0))) / ((g1 . t0) ^2 ))),(r * ((((diff f2,t0) * (g2 . t0)) - ((diff g2,t0) * (f2 . t0))) / ((g2 . t0) ^2 ))),(r * ((((diff f3,t0) * (g3 . t0)) - ((diff g3,t0) * (f3 . t0))) / ((g3 . t0) ^2 )))]|
by XCMPLX_1:75
.=
r * |[((((diff f1,t0) * (g1 . t0)) - ((diff g1,t0) * (f1 . t0))) / ((g1 . t0) ^2 )),((((diff f2,t0) * (g2 . t0)) - ((diff g2,t0) * (f2 . t0))) / ((g2 . t0) ^2 )),((((diff f3,t0) * (g3 . t0)) - ((diff g3,t0) * (f3 . t0))) / ((g3 . t0) ^2 ))]|
by LmA8
;
hence
VFuncdiff ((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0 = r * |[((((diff f1,t0) * (g1 . t0)) - ((diff g1,t0) * (f1 . t0))) / ((g1 . t0) ^2 )),((((diff f2,t0) * (g2 . t0)) - ((diff g2,t0) * (f2 . t0))) / ((g2 . t0) ^2 )),((((diff f3,t0) * (g3 . t0)) - ((diff g3,t0) * (f3 . t0))) / ((g3 . t0) ^2 ))]|
; verum