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`21_in z0 holds
( r (#) (pdiff1 f,2) is_partial_differentiable_in z0,1 & partdiff (r (#) (pdiff1 f,2)),z0,1 = r * (hpartdiff21 f,z0) )
let z0 be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`21_in z0 holds
( r (#) (pdiff1 f,2) is_partial_differentiable_in z0,1 & partdiff (r (#) (pdiff1 f,2)),z0,1 = r * (hpartdiff21 f,z0) )
let f be PartFunc of (REAL 2),REAL ; :: thesis: ( f is_hpartial_differentiable`21_in z0 implies ( r (#) (pdiff1 f,2) is_partial_differentiable_in z0,1 & partdiff (r (#) (pdiff1 f,2)),z0,1 = r * (hpartdiff21 f,z0) ) )
assume A1: f is_hpartial_differentiable`21_in z0 ; :: thesis: ( r (#) (pdiff1 f,2) is_partial_differentiable_in z0,1 & partdiff (r (#) (pdiff1 f,2)),z0,1 = r * (hpartdiff21 f,z0) )
then A2: pdiff1 f,2 is_partial_differentiable_in z0,1 by Th11;
then partdiff (r (#) (pdiff1 f,2)),z0,1 = r * (partdiff (pdiff1 f,2),z0,1) by PDIFF_1:33;
hence ( r (#) (pdiff1 f,2) is_partial_differentiable_in z0,1 & partdiff (r (#) (pdiff1 f,2)),z0,1 = r * (hpartdiff21 f,z0) ) by A1, A2, Th15, PDIFF_1:33; :: thesis: verum