let X, Y be RealLinearSpace; :: thesis: for f, g, h being VECTOR of (R_VectorSpace_of_LinearOperators 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_LinearOperators X,Y); :: thesis: ( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )
reconsider f' = f, g' = g, h' = h as LinearOperator of X,Y by Def7;
A1: R_VectorSpace_of_LinearOperators X,Y is Subspace of RealVectSpace the carrier of X,Y by Th17, RSSPACE:13;
then reconsider f1 = f as VECTOR of (RealVectSpace the carrier of X,Y) by RLSUB_1:18;
reconsider g1 = g as VECTOR of (RealVectSpace the carrier of X,Y) by A1, RLSUB_1:18;
reconsider h1 = h as VECTOR of (RealVectSpace the carrier of X,Y) by A1, RLSUB_1:18;

A2: now

assume A3: 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, A3, RLSUB_1:21;
hence h' . x = (f' . x) + (g' . x) by Th3; :: 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 Th3;
hence h = f + g by A1, RLSUB_1:21; :: thesis: verum

end;

hence ( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) ) by A2; :: thesis: verum