let r be Real; :: thesis: for z0 being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`22_in z0 holds
( r (#) (pdiff1 (f,2)) is_partial_differentiable_in z0,2 & partdiff ((r (#) (pdiff1 (f,2))),z0,2) = r * (hpartdiff22 (f,z0)) )
let z0 be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`22_in z0 holds
( r (#) (pdiff1 (f,2)) is_partial_differentiable_in z0,2 & partdiff ((r (#) (pdiff1 (f,2))),z0,2) = r * (hpartdiff22 (f,z0)) )
let f be PartFunc of (REAL 2),REAL; :: thesis: ( f is_hpartial_differentiable`22_in z0 implies ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in z0,2 & partdiff ((r (#) (pdiff1 (f,2))),z0,2) = r * (hpartdiff22 (f,z0)) ) )
assume A1: f is_hpartial_differentiable`22_in z0 ; :: thesis: ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in z0,2 & partdiff ((r (#) (pdiff1 (f,2))),z0,2) = r * (hpartdiff22 (f,z0)) )
then A2: pdiff1 (f,2) is_partial_differentiable_in z0,2 by Th12;
reconsider r = r as Real ;
partdiff ((r (#) (pdiff1 (f,2))),z0,2) = r * (partdiff ((pdiff1 (f,2)),z0,2)) by PDIFF_1:33, A2;
hence ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in z0,2 & partdiff ((r (#) (pdiff1 (f,2))),z0,2) = r * (hpartdiff22 (f,z0)) ) by A1, A2, Th16, PDIFF_1:33; :: thesis: verum