let r be Real; for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`22_in u0 holds
( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,2))),u0,2) = r * (hpartdiff22 (f,u0)) )
let u0 be Element of REAL 3; for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`22_in u0 holds
( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,2))),u0,2) = r * (hpartdiff22 (f,u0)) )
let f be PartFunc of (REAL 3),REAL; ( f is_hpartial_differentiable`22_in u0 implies ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,2))),u0,2) = r * (hpartdiff22 (f,u0)) ) )
assume A1:
f is_hpartial_differentiable`22_in u0
; ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,2))),u0,2) = r * (hpartdiff22 (f,u0)) )
then
pdiff1 (f,2) is_partial_differentiable_in u0,2
by Th23;
then
( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,2))),u0,2) = r * (partdiff ((pdiff1 (f,2)),u0,2)) )
by PDIFF_1:33;
hence
( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,2))),u0,2) = r * (hpartdiff22 (f,u0)) )
by A1, Th32; verum