let u be Element of REAL 3; for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`22_in u holds
hpartdiff22 (f,u) = partdiff ((pdiff1 (f,2)),u,2)
let f be PartFunc of (REAL 3),REAL; ( f is_hpartial_differentiable`22_in u implies hpartdiff22 (f,u) = partdiff ((pdiff1 (f,2)),u,2) )
assume A1:
f is_hpartial_differentiable`22_in u
; hpartdiff22 (f,u) = partdiff ((pdiff1 (f,2)),u,2)
consider x0, y0, z0 being Element of REAL such that
A2:
u = <*x0,y0,z0*>
by FINSEQ_2:103;
hpartdiff22 (f,u) =
diff ((SVF1 (2,(pdiff1 (f,2)),u)),y0)
by A1, A2, Th14
.=
partdiff ((pdiff1 (f,2)),u,2)
by A2, PDIFF_4:20
;
hence
hpartdiff22 (f,u) = partdiff ((pdiff1 (f,2)),u,2)
; verum