let m, n be non zero Nat; :: thesis: for i, j being Nat

for f being PartFunc of (REAL-NS m),(REAL-NS n)

for h being PartFunc of (REAL m),(REAL n)

for x being Point of (REAL-NS m)

for z being Element of REAL m st f = h & x = z holds

( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j )

let i, j be Nat; :: thesis: for f being PartFunc of (REAL-NS m),(REAL-NS n)

for h being PartFunc of (REAL m),(REAL n)

for x being Point of (REAL-NS m)

for z being Element of REAL m st f = h & x = z holds

( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j )

let f be PartFunc of (REAL-NS m),(REAL-NS n); :: thesis: for h being PartFunc of (REAL m),(REAL n)

for x being Point of (REAL-NS m)

for z being Element of REAL m st f = h & x = z holds

( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j )

let h be PartFunc of (REAL m),(REAL n); :: thesis: for x being Point of (REAL-NS m)

for z being Element of REAL m st f = h & x = z holds

( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j )

let x be Point of (REAL-NS m); :: thesis: for z being Element of REAL m st f = h & x = z holds

( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j )

let z be Element of REAL m; :: thesis: ( f = h & x = z implies ( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j ) )

assume that

A1: f = h and

A2: x = z ; :: thesis: ( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j )

A3: (Proj (i,m)) . x = <*((proj (i,m)) . z)*> by A2, Def4;

((Proj (j,n)) * f) * (reproj (i,x)) = <>* (((proj (j,n)) * h) * (reproj (i,z))) by A1, A2, Th22;

hence ( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j ) by A3, Th7, Th8; :: thesis: verum

for f being PartFunc of (REAL-NS m),(REAL-NS n)

for h being PartFunc of (REAL m),(REAL n)

for x being Point of (REAL-NS m)

for z being Element of REAL m st f = h & x = z holds

( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j )

let i, j be Nat; :: thesis: for f being PartFunc of (REAL-NS m),(REAL-NS n)

for h being PartFunc of (REAL m),(REAL n)

for x being Point of (REAL-NS m)

for z being Element of REAL m st f = h & x = z holds

( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j )

let f be PartFunc of (REAL-NS m),(REAL-NS n); :: thesis: for h being PartFunc of (REAL m),(REAL n)

for x being Point of (REAL-NS m)

for z being Element of REAL m st f = h & x = z holds

( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j )

let h be PartFunc of (REAL m),(REAL n); :: thesis: for x being Point of (REAL-NS m)

for z being Element of REAL m st f = h & x = z holds

( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j )

let x be Point of (REAL-NS m); :: thesis: for z being Element of REAL m st f = h & x = z holds

( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j )

let z be Element of REAL m; :: thesis: ( f = h & x = z implies ( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j ) )

assume that

A1: f = h and

A2: x = z ; :: thesis: ( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j )

A3: (Proj (i,m)) . x = <*((proj (i,m)) . z)*> by A2, Def4;

((Proj (j,n)) * f) * (reproj (i,x)) = <>* (((proj (j,n)) * h) * (reproj (i,z))) by A1, A2, Th22;

hence ( f is_partial_differentiable_in x,i,j iff h is_partial_differentiable_in z,i,j ) by A3, Th7, Th8; :: thesis: verum