let z be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`22_in z holds
hpartdiff22 (f,z) = partdiff ((pdiff1 (f,2)),z,2)
let f be PartFunc of (REAL 2),REAL; :: thesis: ( f is_hpartial_differentiable`22_in z implies hpartdiff22 (f,z) = partdiff ((pdiff1 (f,2)),z,2) )
consider x0, y0 being Element of REAL such that
A1: z = <*x0,y0*> by FINSEQ_2:100;
assume A2: f is_hpartial_differentiable`22_in z ; :: thesis: hpartdiff22 (f,z) = partdiff ((pdiff1 (f,2)),z,2)
then A3: pdiff1 (f,2) is_partial_differentiable_in z,2 by Th12;
hpartdiff22 (f,z) = diff ((SVF1 (2,(pdiff1 (f,2)),z)),y0) by A2, A1, Th8
.= partdiff ((pdiff1 (f,2)),z,2) by A1, A3, PDIFF_2:14 ;
hence hpartdiff22 (f,z) = partdiff ((pdiff1 (f,2)),z,2) ; :: thesis: verum