let u be Element of REAL 3; for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`23_in u holds
hpartdiff23 (f,u) = partdiff ((pdiff1 (f,2)),u,3)
let f be PartFunc of (REAL 3),REAL; ( f is_hpartial_differentiable`23_in u implies hpartdiff23 (f,u) = partdiff ((pdiff1 (f,2)),u,3) )
assume A1:
f is_hpartial_differentiable`23_in u
; hpartdiff23 (f,u) = partdiff ((pdiff1 (f,2)),u,3)
consider x0, y0, z0 being Element of REAL such that
A2:
u = <*x0,y0,z0*>
by FINSEQ_2:103;
hpartdiff23 (f,u) =
diff ((SVF1 (3,(pdiff1 (f,2)),u)),z0)
by A1, A2, Th15
.=
partdiff ((pdiff1 (f,2)),u,3)
by A2, PDIFF_4:21
;
hence
hpartdiff23 (f,u) = partdiff ((pdiff1 (f,2)),u,3)
; verum