let X, Y be ComplexLinearSpace; :: thesis: for f, h being VECTOR of (C_VectorSpace_of_LinearOperators X,Y)
for c being Complex holds
( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
let f, h be VECTOR of (C_VectorSpace_of_LinearOperators X,Y); :: thesis: for c being Complex holds
( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
let c be Complex; :: thesis: ( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
reconsider f' = f, h' = h as LinearOperator of X,Y by Def5;
A1: C_VectorSpace_of_LinearOperators X,Y is Subspace of ComplexVectSpace the carrier of X,Y by Th15, CSSPACE:13;
then reconsider f1 = f as VECTOR of (ComplexVectSpace the carrier of X,Y) by CLVECT_1:30;
reconsider h1 = h as VECTOR of (ComplexVectSpace the carrier of X,Y) by A1, CLVECT_1:30;

A2: now

assume A3: h = c * f ; :: thesis: for x being Element of X holds h' . x = c * (f' . x)
let x be Element of X; :: thesis: h' . x = c * (f' . x)
h1 = c * f1 by A1, A3, CLVECT_1:34;
hence h' . x = c * (f' . x) by Th13; :: thesis: verum

end;

now

assume for x being Element of X holds h' . x = c * (f' . x) ; :: thesis: h = c * f
then h1 = c * f1 by Th13;
hence h = c * f by A1, CLVECT_1:34; :: thesis: verum

end;

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