let S be non empty TopSpace; :: thesis: for T being LinearTopSpace
for f, g, h being VECTOR of (R_VectorSpace_of_ContinuousFunctions (S,T)) holds
( h = f + g iff for x being Element of S holds h . x = (f . x) + (g . x) )
let T be LinearTopSpace; :: thesis: for f, g, h being VECTOR of (R_VectorSpace_of_ContinuousFunctions (S,T)) holds
( h = f + g iff for x being Element of S holds h . x = (f . x) + (g . x) )
let f, g, h be VECTOR of (R_VectorSpace_of_ContinuousFunctions (S,T)); :: thesis: ( h = f + g iff for x being Element of S holds h . x = (f . x) + (g . x) )
set G = { f where f is Function of the carrier of S, the carrier of T : f is continuous } ;
A1: ( f in { f where f is Function of the carrier of S, the carrier of T : f is continuous } & g in { f where f is Function of the carrier of S, the carrier of T : f is continuous } & h in { f where f is Function of the carrier of S, the carrier of T : f is continuous } ) ;
then ex x being Function of the carrier of S, the carrier of T st
( f = x & x is continuous ) ;
then reconsider f9 = f as Function of S,T ;
ex x being Function of the carrier of S, the carrier of T st
( g = x & x is continuous ) by A1;
then reconsider g9 = g as Function of S,T ;
ex x being Function of the carrier of S, the carrier of T st
( h = x & x is continuous ) by A1;
then reconsider h9 = h as Function of S,T ;
A2: R_VectorSpace_of_ContinuousFunctions (S,T) is Subspace of RealVectSpace ( the carrier of S,T) by Th5, RSSPACE:11;
then reconsider f1 = f as VECTOR of (RealVectSpace ( the carrier of S,T)) by RLSUB_1:10;
reconsider h1 = h as VECTOR of (RealVectSpace ( the carrier of S,T)) by A2, RLSUB_1:10;
reconsider g1 = g as VECTOR of (RealVectSpace ( the carrier of S,T)) by A2, RLSUB_1:10;

A3: now :: thesis: ( h = f + g implies for x being Element of S holds h9 . x = (f9 . x) + (g9 . x) )

assume A4: h = f + g ; :: thesis: for x being Element of S holds h9 . x = (f9 . x) + (g9 . x)
let x be Element of S; :: thesis: h9 . x = (f9 . x) + (g9 . x)
h1 = f1 + g1 by A2, A4, RLSUB_1:13;
hence h9 . x = (f9 . x) + (g9 . x) by LOPBAN_1:1; :: thesis: verum

end;

now :: thesis: ( ( for x being Element of S holds h9 . x = (f9 . x) + (g9 . x) ) implies h = f + g )

assume for x being Element of S holds h9 . x = (f9 . x) + (g9 . x) ; :: thesis: h = f + g
then h1 = f1 + g1 by LOPBAN_1:1;
hence h = f + g by A2, RLSUB_1:13; :: thesis: verum

end;

hence ( h = f + g iff for x being Element of S holds h . x = (f . x) + (g . x) ) by A3; :: thesis: verum