let X, Y be ComplexNormSpace; :: thesis: for f, h being VECTOR of (C_VectorSpace_of_BoundedLinearOperators (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_BoundedLinearOperators (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) )
A1: C_VectorSpace_of_BoundedLinearOperators (X,Y) is Subspace of C_VectorSpace_of_LinearOperators (X,Y) by Th21, CSSPACE:11;
then reconsider f1 = f as VECTOR of (C_VectorSpace_of_LinearOperators (X,Y)) by CLVECT_1:29;
reconsider h1 = h as VECTOR of (C_VectorSpace_of_LinearOperators (X,Y)) by A1, CLVECT_1:29;

hereby :: thesis: ( ( for x being VECTOR of X holds h . x = c * (f . x) ) implies h = c * f )

assume A2: 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, A2, CLVECT_1:33;
hence h . x = c * (f . x) by Th16; :: thesis: verum

end;

assume for x being Element of X holds h . x = c * (f . x) ; :: thesis: h = c * f
then h1 = c * f1 by Th16;
hence h = c * f by A1, CLVECT_1:33; :: thesis: verum