theorem :: EUCLID_8:114

for f1, f2, f3, g1, g2, g3, h1, h2, h3 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 & h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0 holds
VFuncdiff (((h2 (#) ((f1 (#) g2) - (f2 (#) g1))) - (h3 (#) ((f3 (#) g1) - (f1 (#) g3)))),((h3 (#) ((f2 (#) g3) - (f3 (#) g2))) - (h1 (#) ((f1 (#) g2) - (f2 (#) g1)))),((h1 (#) ((f3 (#) g1) - (f1 (#) g3))) - (h2 (#) ((f2 (#) g3) - (f3 (#) g2)))),t0) = (|[(((h2 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0))))) - ((h3 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0)))))),(((h3 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))) - ((h1 . t0) * (((f1 . t0) * (diff (g2,t0))) - ((f2 . t0) * (diff (g1,t0)))))),(((h1 . t0) * (((f3 . t0) * (diff (g1,t0))) - ((f1 . t0) * (diff (g3,t0))))) - ((h2 . t0) * (((f2 . t0) * (diff (g3,t0))) - ((f3 . t0) * (diff (g2,t0))))))]| + |[(((h2 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0)))) - ((h3 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0))))),(((h3 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))) - ((h1 . t0) * (((diff (f1,t0)) * (g2 . t0)) - ((diff (f2,t0)) * (g1 . t0))))),(((h1 . t0) * (((diff (f3,t0)) * (g1 . t0)) - ((diff (f1,t0)) * (g3 . t0)))) - ((h2 . t0) * (((diff (f2,t0)) * (g3 . t0)) - ((diff (f3,t0)) * (g2 . t0)))))]|) + |[(((diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - ((diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))),(((diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - ((diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))),(((diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - ((diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]|