let u be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`31_in u holds
hpartdiff31 f,u = partdiff (pdiff1 f,3),u,1
let f be PartFunc of (REAL 3),REAL ; :: thesis: ( f is_hpartial_differentiable`31_in u implies hpartdiff31 f,u = partdiff (pdiff1 f,3),u,1 )
assume A1: f is_hpartial_differentiable`31_in u ; :: thesis: hpartdiff31 f,u = partdiff (pdiff1 f,3),u,1
consider x0, y0, z0 being Real such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:123;
hpartdiff31 f,u = diff (SVF1 1,(pdiff1 f,3),u),x0 by A1, A2, Th7ForSecondOrder
.= partdiff (pdiff1 f,3),u,1 by A2, PDIFF_4:19 ;
hence hpartdiff31 f,u = partdiff (pdiff1 f,3),u,1 ; :: thesis: verum