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 Th43
.= |[((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 Lm6
.= (grad f,p) + |[(partdiff g,p,1),(partdiff g,p,2),(partdiff g,p,3)]| by Th43
.= (grad f,p) + (grad g,p) by Th43 ;
hence grad (f + g),p = (grad f,p) + (grad g,p) ; :: thesis: verum