let X be non empty set ; :: thesis: for Y being RealNormSpace
for f, g, h being VECTOR of (R_VectorSpace_of_BoundedFunctions X,Y)
for f', g', h' being bounded Function of X,the carrier of Y st f' = f & g' = g & h' = h holds
( h = f + g iff for x being Element of X holds h' . x = (f' . x) + (g' . x) )
let Y be RealNormSpace; :: thesis: for f, g, h being VECTOR of (R_VectorSpace_of_BoundedFunctions X,Y)
for f', g', h' being bounded Function of X,the carrier of Y st f' = f & g' = g & h' = h holds
( h = f + g iff for x being Element of X holds h' . x = (f' . x) + (g' . x) )
let f, g, h be VECTOR of (R_VectorSpace_of_BoundedFunctions X,Y); :: thesis: for f', g', h' being bounded Function of X,the carrier of Y st f' = f & g' = g & h' = h holds
( h = f + g iff for x being Element of X holds h' . x = (f' . x) + (g' . x) )
A1: R_VectorSpace_of_BoundedFunctions X,Y is Subspace of RealVectSpace X,Y by Th7, RSSPACE:13;
then reconsider f1 = f as VECTOR of (RealVectSpace X,Y) by RLSUB_1:18;
reconsider h1 = h as VECTOR of (RealVectSpace X,Y) by A1, RLSUB_1:18;
reconsider g1 = g as VECTOR of (RealVectSpace X,Y) by A1, RLSUB_1:18;
let f', g', h' be bounded Function of X,the carrier of Y; :: thesis: ( f' = f & g' = g & h' = h implies ( h = f + g iff for x being Element of X holds h' . x = (f' . x) + (g' . x) ) )
assume A2: ( f' = f & g' = g & h' = h ) ; :: thesis: ( h = f + g iff for x being Element of X holds h' . x = (f' . x) + (g' . x) )

A3: now

assume A4: h = f + g ; :: thesis: for x being Element of X holds h' . x = (f' . x) + (g' . x)
let x be Element of X; :: thesis: h' . x = (f' . x) + (g' . x)
h1 = f1 + g1 by A1, A4, RLSUB_1:21;
hence h' . x = (f' . x) + (g' . x) by A2, LOPBAN_1:14; :: thesis: verum

end;

now

assume for x being Element of X holds h' . x = (f' . x) + (g' . x) ; :: thesis: h = f + g
then h1 = f1 + g1 by A2, LOPBAN_1:14;
hence h = f + g by A1, RLSUB_1:21; :: thesis: verum

end;

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