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) )
reconsider f9 = f, h9 = h as LinearOperator of X,Y by Def4;
let c be Complex; :: thesis: ( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) )
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;

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

assume A3: h = c * f ; :: thesis: for x being Element of X holds h9 . x = c * (f9 . x)
let x be Element of X; :: thesis: h9 . x = c * (f9 . x)
h1 = c * f1 by A1, A3, CLVECT_1:33;
hence h9 . x = c * (f9 . x) by Th12; :: thesis: verum

end;

now :: thesis: ( ( for x being Element of X holds h9 . x = c * (f9 . x) ) implies h = c * f )

assume for x being Element of X holds h9 . x = c * (f9 . x) ; :: thesis: h = c * f
then h1 = c * f1 by Th12;
hence h = c * f by A1, CLVECT_1:33; :: thesis: verum

end;

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