let X be non empty set ; for Y being RealNormSpace
for f, g, h being VECTOR of (R_VectorSpace_of_BoundedFunctions (X,Y))
for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds
( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )
let Y be RealNormSpace; for f, g, h being VECTOR of (R_VectorSpace_of_BoundedFunctions (X,Y))
for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds
( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )
let f, g, h be VECTOR of (R_VectorSpace_of_BoundedFunctions (X,Y)); for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds
( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )
A1:
R_VectorSpace_of_BoundedFunctions (X,Y) is Subspace of RealVectSpace (X,Y)
by Th6, RSSPACE:11;
then reconsider f1 = f as VECTOR of (RealVectSpace (X,Y)) by RLSUB_1:10;
reconsider h1 = h as VECTOR of (RealVectSpace (X,Y)) by A1, RLSUB_1:10;
reconsider g1 = g as VECTOR of (RealVectSpace (X,Y)) by A1, RLSUB_1:10;
let f9, g9, h9 be bounded Function of X, the carrier of Y; ( f9 = f & g9 = g & h9 = h implies ( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ) )
assume A2:
( f9 = f & g9 = g & h9 = h )
; ( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )
A3:
now ( h = f + g implies for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )end;
now ( ( for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ) implies h = f + g )end;
hence
( h = f + g iff for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )
by A3; verum