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 pdiff1 f,1 is_partial_differentiable_in z0,2 by Th10;
then ( r (#) (pdiff1 f,1) is_partial_differentiable_in z0,2 & 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, Th18; :: thesis: verum