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 (#) (pdiff2 f,z0) is_partial_differentiable`1_in z0 & partdiff1 (r (#) (pdiff2 f,z0)),z0 = 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 (#) (pdiff2 f,z0) is_partial_differentiable`1_in z0 & partdiff1 (r (#) (pdiff2 f,z0)),z0 = r * (hpartdiff21 f,z0) )
let f be PartFunc of REAL 2, REAL ; :: thesis: ( f is_hpartial_differentiable`21_in z0 implies ( r (#) (pdiff2 f,z0) is_partial_differentiable`1_in z0 & partdiff1 (r (#) (pdiff2 f,z0)),z0 = r * (hpartdiff21 f,z0) ) )
assume A1: f is_hpartial_differentiable`21_in z0 ; :: thesis: ( r (#) (pdiff2 f,z0) is_partial_differentiable`1_in z0 & partdiff1 (r (#) (pdiff2 f,z0)),z0 = r * (hpartdiff21 f,z0) )
then pdiff2 f,z0 is_partial_differentiable`1_in z0 by Th11;
then ( r (#) (pdiff2 f,z0) is_partial_differentiable`1_in z0 & partdiff1 (r (#) (pdiff2 f,z0)),z0 = r * (partdiff1 (pdiff2 f,z0),z0) ) by PDIFF_2:21;
hence ( r (#) (pdiff2 f,z0) is_partial_differentiable`1_in z0 & partdiff1 (r (#) (pdiff2 f,z0)),z0 = r * (hpartdiff21 f,z0) ) by A1, Th19; :: thesis: verum