let r be Real; :: thesis: for z0 being Element of REAL 2
for f being PartFunc of REAL 2, REAL st f is_partial_differentiable`2_in z0 holds
( r (#) f is_partial_differentiable`2_in z0 & partdiff2 (r (#) f),z0 = r * (partdiff2 f,z0) )
let z0 be Element of REAL 2; :: thesis: for f being PartFunc of REAL 2, REAL st f is_partial_differentiable`2_in z0 holds
( r (#) f is_partial_differentiable`2_in z0 & partdiff2 (r (#) f),z0 = r * (partdiff2 f,z0) )
let f be PartFunc of REAL 2, REAL ; :: thesis: ( f is_partial_differentiable`2_in z0 implies ( r (#) f is_partial_differentiable`2_in z0 & partdiff2 (r (#) f),z0 = r * (partdiff2 f,z0) ) )
assume f is_partial_differentiable`2_in z0 ; :: thesis: ( r (#) f is_partial_differentiable`2_in z0 & partdiff2 (r (#) f),z0 = r * (partdiff2 f,z0) )
then f is_partial_differentiable_in z0,2 by Lem4;
then ( r (#) f is_partial_differentiable_in z0,2 & partdiff (r (#) f),z0,2 = r * (partdiff f,z0,2) ) by PDIFF_1:33;
hence ( r (#) f is_partial_differentiable`2_in z0 & partdiff2 (r (#) f),z0 = r * (partdiff2 f,z0) ) by Lem4; :: thesis: verum