let z be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`11_in z holds
hpartdiff11 (f,z) = partdiff ((pdiff1 (f,1)),z,1)
let f be PartFunc of (REAL 2),REAL; :: thesis: ( f is_hpartial_differentiable`11_in z implies hpartdiff11 (f,z) = partdiff ((pdiff1 (f,1)),z,1) )
consider x0, y0 being Element of REAL such that
A1: z = <*x0,y0*> by FINSEQ_2:100;
assume A2: f is_hpartial_differentiable`11_in z ; :: thesis: hpartdiff11 (f,z) = partdiff ((pdiff1 (f,1)),z,1)
then A3: pdiff1 (f,1) is_partial_differentiable_in z,1 by Th9;
hpartdiff11 (f,z) = diff ((SVF1 (1,(pdiff1 (f,1)),z)),x0) by A2, A1, Th5
.= partdiff ((pdiff1 (f,1)),z,1) by A1, A3, PDIFF_2:13 ;
hence hpartdiff11 (f,z) = partdiff ((pdiff1 (f,1)),z,1) ; :: thesis: verum