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 f9 = f, g9 = g, h9 = 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:11;
then reconsider f1 = f as VECTOR of (RealVectSpace ( the carrier of X,Y)) by RLSUB_1:10;
reconsider h1 = h as VECTOR of (RealVectSpace ( the carrier of X,Y)) by A1, RLSUB_1:10;
reconsider g1 = g as VECTOR of (RealVectSpace ( the carrier of X,Y)) by A1, RLSUB_1:10;

A2: now

assume A3: h = f + g ; :: thesis: for x being Element of X holds h9 . x = (f9 . x) + (g9 . x)
let x be Element of X; :: thesis: h9 . x = (f9 . x) + (g9 . x)
h1 = f1 + g1 by A1, A3, RLSUB_1:13;
hence h9 . x = (f9 . x) + (g9 . x) by Th3; :: thesis: verum

end;

now

assume for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ; :: thesis: h = f + g
then h1 = f1 + g1 by Th3;
hence h = f + g by A1, RLSUB_1:13; :: 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