let f be PartFunc of (REAL 2),REAL ; :: thesis: for z0 being Element of REAL 2
for N being Neighbourhood of (proj 2,2) . z0 st f is_hpartial_differentiable`22_in z0 & N c= dom (SVF1 2,(pdiff1 f,2),z0) holds
for h being convergent_to_0 Real_Sequence
for c being V20() Real_Sequence st rng c = {((proj 2,2) . z0)} & rng (h + c) c= N holds
( (h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c)) is convergent & hpartdiff22 f,z0 = lim ((h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c))) )
let z0 be Element of REAL 2; :: thesis: for N being Neighbourhood of (proj 2,2) . z0 st f is_hpartial_differentiable`22_in z0 & N c= dom (SVF1 2,(pdiff1 f,2),z0) holds
for h being convergent_to_0 Real_Sequence
for c being V20() Real_Sequence st rng c = {((proj 2,2) . z0)} & rng (h + c) c= N holds
( (h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c)) is convergent & hpartdiff22 f,z0 = lim ((h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c))) )
let N be Neighbourhood of (proj 2,2) . z0; :: thesis: ( f is_hpartial_differentiable`22_in z0 & N c= dom (SVF1 2,(pdiff1 f,2),z0) implies for h being convergent_to_0 Real_Sequence
for c being V20() Real_Sequence st rng c = {((proj 2,2) . z0)} & rng (h + c) c= N holds
( (h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c)) is convergent & hpartdiff22 f,z0 = lim ((h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c))) ) )
assume that
A1: f is_hpartial_differentiable`22_in z0 and
A2: N c= dom (SVF1 2,(pdiff1 f,2),z0) ; :: thesis: for h being convergent_to_0 Real_Sequence
for c being V20() Real_Sequence st rng c = {((proj 2,2) . z0)} & rng (h + c) c= N holds
( (h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c)) is convergent & hpartdiff22 f,z0 = lim ((h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c))) )
let h be convergent_to_0 Real_Sequence; :: thesis: for c being V20() Real_Sequence st rng c = {((proj 2,2) . z0)} & rng (h + c) c= N holds
( (h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c)) is convergent & hpartdiff22 f,z0 = lim ((h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c))) )
let c be V20() Real_Sequence; :: thesis: ( rng c = {((proj 2,2) . z0)} & rng (h + c) c= N implies ( (h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c)) is convergent & hpartdiff22 f,z0 = lim ((h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c))) ) )
assume A3: ( rng c = {((proj 2,2) . z0)} & rng (h + c) c= N ) ; :: thesis: ( (h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c)) is convergent & hpartdiff22 f,z0 = lim ((h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c))) )
z0 is Tuple of 2, REAL by FINSEQ_2:151;
then consider x0, y0 being Real such that
A4: z0 = <*x0,y0*> by FINSEQ_2:120;
A5: pdiff1 f,2 is_partial_differentiable_in z0,2 by A1, Th12;
then partdiff (pdiff1 f,2),z0,2 = diff (SVF1 2,(pdiff1 f,2),z0),y0 by A4, PDIFF_2:14
.= hpartdiff22 f,z0 by A1, A4, Th8 ;
hence ( (h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c)) is convergent & hpartdiff22 f,z0 = lim ((h " ) (#) (((SVF1 2,(pdiff1 f,2),z0) /* (h + c)) - ((SVF1 2,(pdiff1 f,2),z0) /* c))) ) by A2, A3, A5, PDIFF_2:18; :: thesis: verum