let r be Real; for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`33_in u0 holds
( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,3))),u0,3) = r * (hpartdiff33 (f,u0)) )
let u0 be Element of REAL 3; for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`33_in u0 holds
( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,3))),u0,3) = r * (hpartdiff33 (f,u0)) )
let f be PartFunc of (REAL 3),REAL; ( f is_hpartial_differentiable`33_in u0 implies ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,3))),u0,3) = r * (hpartdiff33 (f,u0)) ) )
assume A1:
f is_hpartial_differentiable`33_in u0
; ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,3))),u0,3) = r * (hpartdiff33 (f,u0)) )
then
pdiff1 (f,3) is_partial_differentiable_in u0,3
by Th27;
then
( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,3))),u0,3) = r * (partdiff ((pdiff1 (f,3)),u0,3)) )
by PDIFF_1:33;
hence
( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,3))),u0,3) = r * (hpartdiff33 (f,u0)) )
by A1, Th36; verum