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 & f is_partial_differentiable_in x,i,j holds
(partdiff (f,x,i,j)) . <*1*> = <*(partdiff (h,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 & f is_partial_differentiable_in x,i,j holds
(partdiff (f,x,i,j)) . <*1*> = <*(partdiff (h,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 & f is_partial_differentiable_in x,i,j holds
(partdiff (f,x,i,j)) . <*1*> = <*(partdiff (h,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 & f is_partial_differentiable_in x,i,j holds
(partdiff (f,x,i,j)) . <*1*> = <*(partdiff (h,z,i,j))*>
let x be Point of (REAL-NS m); :: thesis: for z being Element of REAL m st f = h & x = z & f is_partial_differentiable_in x,i,j holds
(partdiff (f,x,i,j)) . <*1*> = <*(partdiff (h,z,i,j))*>
let z be Element of REAL m; :: thesis: ( f = h & x = z & f is_partial_differentiable_in x,i,j implies (partdiff (f,x,i,j)) . <*1*> = <*(partdiff (h,z,i,j))*> )
assume that
A1: f = h and
A2: x = z and
A3: f is_partial_differentiable_in x,i,j ; :: thesis: (partdiff (f,x,i,j)) . <*1*> = <*(partdiff (h,z,i,j))*>
A4: (Proj (i,m)) . x = <*((proj (i,m)) . z)*> by A2, Def4;
A5: ((Proj (j,n)) * f) * (reproj (i,x)) is_differentiable_in (Proj (i,m)) . x by A3;
((Proj (j,n)) * f) * (reproj (i,x)) = <>* (((proj (j,n)) * h) * (reproj (i,z))) by A1, A2, Th22;
hence (partdiff (f,x,i,j)) . <*1*> = <*(partdiff (h,z,i,j))*> by A4, A5, Th10; :: thesis: verum