let u be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`32_in u holds
hpartdiff32 f,u = partdiff (pdiff1 f,3),u,2
let f be PartFunc of (REAL 3),REAL ; :: thesis: ( f is_hpartial_differentiable`32_in u implies hpartdiff32 f,u = partdiff (pdiff1 f,3),u,2 )
assume A1: f is_hpartial_differentiable`32_in u ; :: thesis: hpartdiff32 f,u = partdiff (pdiff1 f,3),u,2
consider x0, y0, z0 being Real such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:123;
hpartdiff32 f,u = diff (SVF1 2,(pdiff1 f,3),u),y0 by A1, A2, Th8ForSecondOrder
.= partdiff (pdiff1 f,3),u,2 by A2, PDIFF_4:20 ;
hence hpartdiff32 f,u = partdiff (pdiff1 f,3),u,2 ; :: thesis: verum