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 Th22, RSSPACE:11;
then reconsider f1 = f as VECTOR of (R_VectorSpace_of_LinearOperators (X,Y)) by RLSUB_1:10;
reconsider h1 = h as VECTOR of (R_VectorSpace_of_LinearOperators (X,Y)) by A1, RLSUB_1:10;
reconsider g1 = g as VECTOR of (R_VectorSpace_of_LinearOperators (X,Y)) by A1, RLSUB_1:10;

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_1:13;
hence h . x = (f . x) + (g . x) by Th16; :: 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 Th16;
hence h = f + g by A1, RLSUB_1:13; :: thesis: verum