let X, Y be ComplexLinearSpace; :: thesis: for f, g, h being VECTOR of (C_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 (C_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 Def4;
A1: C_VectorSpace_of_LinearOperators (X,Y) is Subspace of ComplexVectSpace ( the carrier of X,Y) by Th13, CSSPACE:11;
then reconsider f1 = f as VECTOR of (ComplexVectSpace ( the carrier of X,Y)) by CLVECT_1:29;
reconsider h1 = h as VECTOR of (ComplexVectSpace ( the carrier of X,Y)) by A1, CLVECT_1:29;
reconsider g1 = g as VECTOR of (ComplexVectSpace ( the carrier of X,Y)) by A1, CLVECT_1:29;

A2: now :: thesis: ( h = f + g implies for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) )

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, CLVECT_1:32;
hence h9 . x = (f9 . x) + (g9 . x) by LOPBAN_1:1; :: thesis: verum

end;

now :: thesis: ( ( for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ) implies h = f + g )

assume for x being Element of X holds h9 . x = (f9 . x) + (g9 . x) ; :: thesis: h = f + g
then h1 = f1 + g1 by LOPBAN_1:1;
hence h = f + g by A1, CLVECT_1:32; :: 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