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`12_in z0 holds
( r (#) (pdiff1 (f,1)) is_partial_differentiable_in z0,2 & partdiff ((r (#) (pdiff1 (f,1))),z0,2) = r * (hpartdiff12 (f,z0)) )
let z0 be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`12_in z0 holds
( r (#) (pdiff1 (f,1)) is_partial_differentiable_in z0,2 & partdiff ((r (#) (pdiff1 (f,1))),z0,2) = r * (hpartdiff12 (f,z0)) )
let f be PartFunc of (REAL 2),REAL; :: thesis: ( f is_hpartial_differentiable`12_in z0 implies ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in z0,2 & partdiff ((r (#) (pdiff1 (f,1))),z0,2) = r * (hpartdiff12 (f,z0)) ) )
assume A1: f is_hpartial_differentiable`12_in z0 ; :: thesis: ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in z0,2 & partdiff ((r (#) (pdiff1 (f,1))),z0,2) = r * (hpartdiff12 (f,z0)) )
then A2: pdiff1 (f,1) is_partial_differentiable_in z0,2 by Th10;
then partdiff ((r (#) (pdiff1 (f,1))),z0,2) = r * (partdiff ((pdiff1 (f,1)),z0,2)) by PDIFF_1:33;
hence ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in z0,2 & partdiff ((r (#) (pdiff1 (f,1))),z0,2) = r * (hpartdiff12 (f,z0)) ) by A1, A2, Th14, PDIFF_1:33; :: thesis: verum