let X, Y be RealNormSpace; :: thesis: for Z being non empty open Subset of X
for n being non empty Nat
for f, g, h being VECTOR of (RVectSp_Continuously_differentiable_Funcs (Z,Y,n)) holds
( h = f + g iff for x being Element of Z holds h . x = (f . x) + (g . x) )
let Z be non empty open Subset of X; :: thesis: for n being non empty Nat
for f, g, h being VECTOR of (RVectSp_Continuously_differentiable_Funcs (Z,Y,n)) holds
( h = f + g iff for x being Element of Z holds h . x = (f . x) + (g . x) )
let n be non empty Nat; :: thesis: for f, g, h being VECTOR of (RVectSp_Continuously_differentiable_Funcs (Z,Y,n)) holds
( h = f + g iff for x being Element of Z holds h . x = (f . x) + (g . x) )
let f, g, h be VECTOR of (RVectSp_Continuously_differentiable_Funcs (Z,Y,n)); :: thesis: ( h = f + g iff for x being Element of Z holds h . x = (f . x) + (g . x) )
A1: RVectSp_Continuously_differentiable_Funcs (Z,Y,n) is Subspace of RealVectSpace (Z,Y) by Th5, RSSPACE:11;
then reconsider f1 = f as VECTOR of (RealVectSpace (Z,Y)) by RLSUB_1:10;
reconsider h1 = h as VECTOR of (RealVectSpace (Z,Y)) by A1, RLSUB_1:10;
reconsider g1 = g as VECTOR of (RealVectSpace (Z,Y)) by A1, RLSUB_1:10;

A2: now :: thesis: ( h = f + g implies for x being Element of Z holds h . x = (f . x) + (g . x) )

assume A3: h = f + g ; :: thesis: for x being Element of Z holds h . x = (f . x) + (g . x)
let x be Element of Z; :: thesis: h . x = (f . x) + (g . x)
h1 = f1 + g1 by A1, A3, RLSUB_1:13;
hence h . x = (f . x) + (g . x) by LOPBAN_1:11; :: thesis: verum

end;

now :: thesis: ( ( for x being Element of Z holds h . x = (f . x) + (g . x) ) implies h = f + g )

assume for x being Element of Z holds h . x = (f . x) + (g . x) ; :: thesis: h = f + g
then h1 = f1 + g1 by LOPBAN_1:11;
hence h = f + g by A1, RLSUB_1:13; :: thesis: verum

end;

hence ( h = f + g iff for x being Element of Z holds h . x = (f . x) + (g . x) ) by A2; :: thesis: verum