let X, Y be ComplexLinearSpace; :: thesis: for f, h being VECTOR of ()
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 (); :: 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 ;
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 ;
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 ;
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 ; :: thesis: verum
end;
hence ( h = c * f iff for x being VECTOR of X holds h . x = c * (f . x) ) by A2; :: thesis: verum