let r be Real; for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`11_in u0 holds
( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,1))),u0,1) = r * (hpartdiff11 (f,u0)) )
let u0 be Element of REAL 3; for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`11_in u0 holds
( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,1))),u0,1) = r * (hpartdiff11 (f,u0)) )
let f be PartFunc of (REAL 3),REAL; ( f is_hpartial_differentiable`11_in u0 implies ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,1))),u0,1) = r * (hpartdiff11 (f,u0)) ) )
assume A1:
f is_hpartial_differentiable`11_in u0
; ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,1))),u0,1) = r * (hpartdiff11 (f,u0)) )
then
pdiff1 (f,1) is_partial_differentiable_in u0,1
by Th19;
then
( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,1))),u0,1) = r * (partdiff ((pdiff1 (f,1)),u0,1)) )
by PDIFF_1:33;
hence
( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,1))),u0,1) = r * (hpartdiff11 (f,u0)) )
by A1, Th28; verum