let p be Element of REAL 3; :: thesis: for f, g being PartFunc of (REAL 3),REAL st f is_partial_differentiable_in p,1 & f is_partial_differentiable_in p,2 & f is_partial_differentiable_in p,3 & g is_partial_differentiable_in p,1 & g is_partial_differentiable_in p,2 & g is_partial_differentiable_in p,3 holds
grad ((f + g),p) = (grad (f,p)) + (grad (g,p))
let f, g be PartFunc of (REAL 3),REAL; :: thesis: ( f is_partial_differentiable_in p,1 & f is_partial_differentiable_in p,2 & f is_partial_differentiable_in p,3 & g is_partial_differentiable_in p,1 & g is_partial_differentiable_in p,2 & g is_partial_differentiable_in p,3 implies grad ((f + g),p) = (grad (f,p)) + (grad (g,p)) )
assume that
A1: ( f is_partial_differentiable_in p,1 & f is_partial_differentiable_in p,2 & f is_partial_differentiable_in p,3 ) and
A2: ( g is_partial_differentiable_in p,1 & g is_partial_differentiable_in p,2 & g is_partial_differentiable_in p,3 ) ; :: thesis: grad ((f + g),p) = (grad (f,p)) + (grad (g,p))
grad ((f + g),p) = |[(partdiff ((f + g),p,1)),(partdiff ((f + g),p,2)),(partdiff ((f + g),p,3))]| by Th34
.= |[((partdiff (f,p,1)) + (partdiff (g,p,1))),(partdiff ((f + g),p,2)),(partdiff ((f + g),p,3))]| by A1, A2, PDIFF_1:29
.= |[((partdiff (f,p,1)) + (partdiff (g,p,1))),((partdiff (f,p,2)) + (partdiff (g,p,2))),(partdiff ((f + g),p,3))]| by A1, A2, PDIFF_1:29
.= |[((partdiff (f,p,1)) + (partdiff (g,p,1))),((partdiff (f,p,2)) + (partdiff (g,p,2))),((partdiff (f,p,3)) + (partdiff (g,p,3)))]| by A1, A2, PDIFF_1:29
.= |[(partdiff (f,p,1)),(partdiff (f,p,2)),(partdiff (f,p,3))]| + |[(partdiff (g,p,1)),(partdiff (g,p,2)),(partdiff (g,p,3))]| by Lm4
.= (grad (f,p)) + |[(partdiff (g,p,1)),(partdiff (g,p,2)),(partdiff (g,p,3))]| by Th34
.= (grad (f,p)) + (grad (g,p)) by Th34 ;
hence grad ((f + g),p) = (grad (f,p)) + (grad (g,p)) ; :: thesis: verum