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 )
z is Tuple of 2, REAL by FINSEQ_2:151;
then consider x0, y0 being Real such that
A1: z = <*x0,y0*> by FINSEQ_2:120;
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