let i be Element of NAT ; :: thesis: for m, n being non empty Element of NAT
for F being PartFunc of (REAL-NS m),(REAL-NS n)
for G being PartFunc of (REAL m),(REAL n)
for x being Point of (REAL-NS m)
for y being Element of REAL m st F = G & x = y & F is_partial_differentiable_in x,i holds
(partdiff F,x,i) . <*1*> = partdiff G,y,i
let m, n be non empty Element of NAT ; :: thesis: for F being PartFunc of (REAL-NS m),(REAL-NS n)
for G being PartFunc of (REAL m),(REAL n)
for x being Point of (REAL-NS m)
for y being Element of REAL m st F = G & x = y & F is_partial_differentiable_in x,i holds
(partdiff F,x,i) . <*1*> = partdiff G,y,i
let F be PartFunc of (REAL-NS m),(REAL-NS n); :: thesis: for G being PartFunc of (REAL m),(REAL n)
for x being Point of (REAL-NS m)
for y being Element of REAL m st F = G & x = y & F is_partial_differentiable_in x,i holds
(partdiff F,x,i) . <*1*> = partdiff G,y,i
let G be PartFunc of (REAL m),(REAL n); :: thesis: for x being Point of (REAL-NS m)
for y being Element of REAL m st F = G & x = y & F is_partial_differentiable_in x,i holds
(partdiff F,x,i) . <*1*> = partdiff G,y,i
let x be Point of (REAL-NS m); :: thesis: for y being Element of REAL m st F = G & x = y & F is_partial_differentiable_in x,i holds
(partdiff F,x,i) . <*1*> = partdiff G,y,i
let y be Element of REAL m; :: thesis: ( F = G & x = y & F is_partial_differentiable_in x,i implies (partdiff F,x,i) . <*1*> = partdiff G,y,i )
assume that
A1: F = G and
A2: x = y and
A3: F is_partial_differentiable_in x,i ; :: thesis: (partdiff F,x,i) . <*1*> = partdiff G,y,i
A4: the carrier of (REAL-NS n) = REAL n by REAL_NS1:def 4;
the carrier of (REAL-NS 1) = REAL 1 by REAL_NS1:def 4;
then partdiff F,x,i is Function of (REAL 1),(REAL n) by A4, LOPBAN_1:def 10;
then A5: (partdiff F,x,i) . <*1*> is Element of REAL n by FINSEQ_2:118, FUNCT_2:7;
G is_partial_differentiable_in y,i by A1, A2, A3, Th16;
hence (partdiff F,x,i) . <*1*> = partdiff G,y,i by A1, A2, A5, Def14; :: thesis: verum