let u be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`33_in u holds
hpartdiff33 f,u = partdiff (pdiff1 f,3),u,3
let f be PartFunc of (REAL 3),REAL ; :: thesis: ( f is_hpartial_differentiable`33_in u implies hpartdiff33 f,u = partdiff (pdiff1 f,3),u,3 )
assume A1: f is_hpartial_differentiable`33_in u ; :: thesis: hpartdiff33 f,u = partdiff (pdiff1 f,3),u,3
consider x0, y0, z0 being Real such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:123;
hpartdiff33 f,u = diff (SVF1 3,(pdiff1 f,3),u),z0 by A1, A2, Th8ForZForSecondOrder
.= partdiff (pdiff1 f,3),u,3 by A2, PDIFF_4:21 ;
hence hpartdiff33 f,u = partdiff (pdiff1 f,3),u,3 ; :: thesis: verum