let u be Element of REAL 3; :: thesis: 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; :: thesis: ( 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 ; :: thesis: 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) ; :: thesis: verum