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 Element of REAL such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
hpartdiff31 (f,u) = diff ((SVF1 (1,(pdiff1 (f,3)),u)),x0) by A1, A2, Th16
.= partdiff ((pdiff1 (f,3)),u,1) by A2, PDIFF_4:19 ;
hence hpartdiff31 (f,u) = partdiff ((pdiff1 (f,3)),u,1) ; :: thesis: verum