let f be PartFunc of (REAL 3),REAL; :: thesis: for u0 being Element of REAL 3
for N being Neighbourhood of (proj (1,3)) . u0 st f is_hpartial_differentiable`21_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,2)),u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c)) is convergent & hpartdiff21 (f,u0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c))) )
let u0 be Element of REAL 3; :: thesis: for N being Neighbourhood of (proj (1,3)) . u0 st f is_hpartial_differentiable`21_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,2)),u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c)) is convergent & hpartdiff21 (f,u0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c))) )
let N be Neighbourhood of (proj (1,3)) . u0; :: thesis: ( f is_hpartial_differentiable`21_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,2)),u0)) implies for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c)) is convergent & hpartdiff21 (f,u0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c))) ) )
assume A1: ( f is_hpartial_differentiable`21_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,2)),u0)) ) ; :: thesis: for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c)) is convergent & hpartdiff21 (f,u0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c))) )
let h be non-zero 0 -convergent Real_Sequence; :: thesis: for c being constant Real_Sequence st rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c)) is convergent & hpartdiff21 (f,u0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c))) )
let c be constant Real_Sequence; :: thesis: ( rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N implies ( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c)) is convergent & hpartdiff21 (f,u0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c))) ) )
assume A2: ( rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N ) ; :: thesis: ( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c)) is convergent & hpartdiff21 (f,u0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c))) )
A3: pdiff1 (f,2) is_partial_differentiable_in u0,1 by A1, Th22;
consider x0, y0, z0 being Element of REAL such that
A4: u0 = <*x0,y0,z0*> by FINSEQ_2:103;
partdiff ((pdiff1 (f,2)),u0,1) = diff ((SVF1 (1,(pdiff1 (f,2)),u0)),x0) by A4, PDIFF_4:19
.= hpartdiff21 (f,u0) by A1, A4, Th13 ;
hence ( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c)) is convergent & hpartdiff21 (f,u0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c))) ) by A1, A2, A3, PDIFF_4:25; :: thesis: verum