let f be PartFunc of (REAL 2),REAL; for z0 being Element of REAL 2
for N being Neighbourhood of (proj (1,2)) . z0 st f is_hpartial_differentiable`21_in z0 & N c= dom (SVF1 (1,(pdiff1 (f,2)),z0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,2)) . z0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c)) is convergent & hpartdiff21 (f,z0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c))) )
let z0 be Element of REAL 2; for N being Neighbourhood of (proj (1,2)) . z0 st f is_hpartial_differentiable`21_in z0 & N c= dom (SVF1 (1,(pdiff1 (f,2)),z0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,2)) . z0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c)) is convergent & hpartdiff21 (f,z0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c))) )
let N be Neighbourhood of (proj (1,2)) . z0; ( f is_hpartial_differentiable`21_in z0 & N c= dom (SVF1 (1,(pdiff1 (f,2)),z0)) implies for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,2)) . z0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c)) is convergent & hpartdiff21 (f,z0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c))) ) )
assume that
A1:
f is_hpartial_differentiable`21_in z0
and
A2:
N c= dom (SVF1 (1,(pdiff1 (f,2)),z0))
; for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,2)) . z0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c)) is convergent & hpartdiff21 (f,z0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c))) )
let h be non-zero 0 -convergent Real_Sequence; for c being constant Real_Sequence st rng c = {((proj (1,2)) . z0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c)) is convergent & hpartdiff21 (f,z0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c))) )
let c be constant Real_Sequence; ( rng c = {((proj (1,2)) . z0)} & rng (h + c) c= N implies ( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c)) is convergent & hpartdiff21 (f,z0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c))) ) )
assume A3:
( rng c = {((proj (1,2)) . z0)} & rng (h + c) c= N )
; ( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c)) is convergent & hpartdiff21 (f,z0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c))) )
consider x0, y0 being Element of REAL such that
A4:
z0 = <*x0,y0*>
by FINSEQ_2:100;
A5:
pdiff1 (f,2) is_partial_differentiable_in z0,1
by A1, Th11;
then partdiff ((pdiff1 (f,2)),z0,1) =
diff ((SVF1 (1,(pdiff1 (f,2)),z0)),x0)
by A4, PDIFF_2:13
.=
hpartdiff21 (f,z0)
by A1, A4, Th7
;
hence
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c)) is convergent & hpartdiff21 (f,z0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c))) )
by A2, A3, A5, PDIFF_2:17; verum