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 t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 holds
VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0) = (VFuncdiff (f1,f2,f3,t0)) + (VFuncdiff (g1,g2,g3,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 t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 implies VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0) = (VFuncdiff (f1,f2,f3,t0)) + (VFuncdiff (g1,g2,g3,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 t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 ) ; :: thesis: VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0) = (VFuncdiff (f1,f2,f3,t0)) + (VFuncdiff (g1,g2,g3,t0))
set p = |[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]|;
set q = |[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]|;
VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0) = |[((diff (f1,t0)) + (diff (g1,t0))),(diff ((f2 + g2),t0)),(diff ((f3 + g3),t0))]| by A1, A2, FDIFF_1:13
.= |[((diff (f1,t0)) + (diff (g1,t0))),((diff (f2,t0)) + (diff (g2,t0))),(diff ((f3 + g3),t0))]| by A1, A2, FDIFF_1:13
.= |[((|[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 1) + (|[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 1)),((|[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 2) + (|[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 2)),((|[(diff (f1,t0)),(diff (f2,t0)),(diff (f3,t0))]| . 3) + (|[(diff (g1,t0)),(diff (g2,t0)),(diff (g3,t0))]| . 3))]| by A1, A2, FDIFF_1:13
.= (VFuncdiff (f1,f2,f3,t0)) + (VFuncdiff (g1,g2,g3,t0)) by Lm2 ;
hence VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0) = (VFuncdiff (f1,f2,f3,t0)) + (VFuncdiff (g1,g2,g3,t0)) ; :: thesis: verum