let X, Y be RealNormSpace; :: thesis: for f, g, h being VECTOR of (R_VectorSpace_of_BoundedLinearOperators X,Y) holds
( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )
let f, g, h be VECTOR of (R_VectorSpace_of_BoundedLinearOperators X,Y); :: thesis: ( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )
A1: R_VectorSpace_of_BoundedLinearOperators X,Y is Subspace of R_VectorSpace_of_LinearOperators X,Y by Th26, RSSPACE:13;
then reconsider f1 = f as VECTOR of (R_VectorSpace_of_LinearOperators X,Y) by RLSUB_1:18;
reconsider g1 = g as VECTOR of (R_VectorSpace_of_LinearOperators X,Y) by A1, RLSUB_1:18;
reconsider h1 = h as VECTOR of (R_VectorSpace_of_LinearOperators X,Y) by A1, RLSUB_1:18;

hereby :: thesis: ( ( for x being VECTOR of X holds h . x = (f . x) + (g . x) ) implies h = f + g )

assume A2: 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, A2, RLSUB_1:21;
hence h . x = (f . x) + (g . x) by Th20; :: thesis: verum

end;

assume for x being Element of X holds h . x = (f . x) + (g . x) ; :: thesis: h = f + g
then h1 = f1 + g1 by Th20;
hence h = f + g by A1, RLSUB_1:21; :: thesis: verum