let m, n be non zero 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_differentiable_in x holds

diff (F,x) = diff (G,y)

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_differentiable_in x holds

diff (F,x) = diff (G,y)

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_differentiable_in x holds

diff (F,x) = diff (G,y)

let x be Point of (REAL-NS m); :: thesis: for y being Element of REAL m st F = G & x = y & F is_differentiable_in x holds

diff (F,x) = diff (G,y)

let y be Element of REAL m; :: thesis: ( F = G & x = y & F is_differentiable_in x implies diff (F,x) = diff (G,y) )

assume that

A1: F = G and

A2: x = y and

A3: F is_differentiable_in x ; :: thesis: diff (F,x) = diff (G,y)

A4: the carrier of (REAL-NS n) = REAL n by REAL_NS1:def 4;

the carrier of (REAL-NS m) = REAL m by REAL_NS1:def 4;

then A5: diff (F,x) is Function of (REAL m),(REAL n) by A4, LOPBAN_1:def 9;

G is_differentiable_in y by A1, A2, A3;

hence diff (F,x) = diff (G,y) by A1, A2, A5, Def8; :: thesis: verum

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_differentiable_in x holds

diff (F,x) = diff (G,y)

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_differentiable_in x holds

diff (F,x) = diff (G,y)

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_differentiable_in x holds

diff (F,x) = diff (G,y)

let x be Point of (REAL-NS m); :: thesis: for y being Element of REAL m st F = G & x = y & F is_differentiable_in x holds

diff (F,x) = diff (G,y)

let y be Element of REAL m; :: thesis: ( F = G & x = y & F is_differentiable_in x implies diff (F,x) = diff (G,y) )

assume that

A1: F = G and

A2: x = y and

A3: F is_differentiable_in x ; :: thesis: diff (F,x) = diff (G,y)

A4: the carrier of (REAL-NS n) = REAL n by REAL_NS1:def 4;

the carrier of (REAL-NS m) = REAL m by REAL_NS1:def 4;

then A5: diff (F,x) is Function of (REAL m),(REAL n) by A4, LOPBAN_1:def 9;

G is_differentiable_in y by A1, A2, A3;

hence diff (F,x) = diff (G,y) by A1, A2, A5, Def8; :: thesis: verum