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 = partdiff1 (pdiff1 f,z),z
let f be PartFunc of REAL 2, REAL ; :: thesis: ( f is_hpartial_differentiable`11_in z implies hpartdiff11 f,z = partdiff1 (pdiff1 f,z),z )
assume A1: f is_hpartial_differentiable`11_in z ; :: thesis: hpartdiff11 f,z = partdiff1 (pdiff1 f,z),z
consider x0, y0 being Real such that
A2: z = <*x0,y0*> by FINSEQ_2:120;
A3: pdiff1 f,z is_partial_differentiable`1_in z by A1, Th9;
hpartdiff11 f,z = diff (SVF1 (pdiff1 f,z),z),x0 by A1, A2, Th5
.= partdiff1 (pdiff1 f,z),z by A2, A3, PDIFF_2:11 ;
hence hpartdiff11 f,z = partdiff1 (pdiff1 f,z),z ; :: thesis: verum