let f1, f2, f3, g1, g2, g3 be PartFunc of REAL,REAL; :: thesis: for t0 being Real st f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in f1 . t0 & g2 is_differentiable_in f2 . t0 & g3 is_differentiable_in f3 . t0 holds
VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0) = |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]|
let t0 be Real; :: thesis: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 & g1 is_differentiable_in f1 . t0 & g2 is_differentiable_in f2 . t0 & g3 is_differentiable_in f3 . t0 implies VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0) = |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| )
assume that
A1: ( f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 ) and
A2: ( g1 is_differentiable_in f1 . t0 & g2 is_differentiable_in f2 . t0 & g3 is_differentiable_in f3 . t0 ) ; :: thesis: VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0) = |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]|
VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0) = |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),(diff ((g2 * f2),t0)),(diff ((g3 * f3),t0))]| by A1, A2, FDIFF_2:13
.= |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),(diff ((g3 * f3),t0))]| by A1, A2, FDIFF_2:13
.= |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| by A1, A2, FDIFF_2:13 ;
hence VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0) = |[((diff (g1,(f1 . t0))) * (diff (f1,t0))),((diff (g2,(f2 . t0))) * (diff (f2,t0))),((diff (g3,(f3 . t0))) * (diff (f3,t0)))]| ; :: thesis: verum